kinetic boltzmann, vlasov and related equations...dulov 03-pref-xi-xiv-9780123877796 2011/5/25 0:41...

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Kinetic Boltzmann, Vlasov and Related Equations

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Kinetic Boltzmann, Vlasovand Related Equations

Victor VedenyapinKeldysh Institute of Applied Mathematics(Russian Academy of Sciences)Russia

Alexander SinitsynDepartamento de MathematicasFacultad de CienciasUniversidad Nacional de ColombiaBogota, Colombia

Eugene DulovFacultad de Ciencia y TecnologıaUniversidad de Ciencias Aplicadas yAmbientales U.D.C.A

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORDPARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

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Preface

The Boltzmann and Vlasov equations played a great role in the past. Their importancecan still be seen in modern natural sciences, technique, engineering, and even in thephilosophy of science. The classical Boltzmann equation derived in 1872 becamea cornerstone for the molecular–kinetic theory, the second law of thermodynamics(increasing of enthropy), and for the derivaion of the basic hydrodynamic equations.Discovering and studying different physical, chemical, and astronomy objects and pro-cesses and even popular nanotechnological applications opened new fields for Boltz-mann and Vlasov equations. Examples include diluted gas, radiation, neutral particlestransportation, atmosphere optics, nuclear reactor modeling, and so on.

The Vlasov equation was obtained in 1938 and served as a basis of plasma physics,but it also describes large-scale processes and galaxies in astronomy (the starwindtheory). The development of plasma units such as tokomak or plasma engines also aresupported by the Vlasov equation.

A careful reader who has looked at the Table of Contents of this book will note thatit contains not only the basics and common facts, but many of the results discussed inthis book were obtained recently.

Hence, the first chapter is devoted to the historical introduction and outlines prin-ciple details described in other chapters. The second chapter introduces Vlasov-typeequations or equations of the self-consistent fields in connection with a problem ofmultiple body dynamics and the use of the Lagrangian coordinates in the Vlasov equa-tion. It also reminds us about its links with hydrodynamical descriptions. For betterunderstanding, we present several examples with exact solutions.

The Vlasov-Maxwell equations are introduced in Chapter 3. To make the derivationtechnique comprehensible, first we start with the particle shift along the trajectories ofan arbitrary system of the ordinary differential equations; we then follow with particlesystem movement in metric spaces. The next chapter deals with the Vlasov equa-tion for plasma and energetic substitution, given with an analogy with the Bernoulliintegral.

In Chapter 5 we introduce kinetic equations, the Boltzmann equation, the Vlasov-Poisson (VP), and the Vlasov-Maxwell (VM) systems, and describe their mathematicalstructure. Section 5.6 describes several open fundamental problems known for VP andVM systems.

Chapter 6 lists references and is devoted to students, engineers, and postgradu-ate students. Here we introduce an ansatz of distribution function for two-componentplasma. Simple problem statements are introduced for nonlinear elliptic equationsboth for Cauchy and bifurcation cases.

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xii Preface

In Chapter 7, we study special cases of stationary and nonstationary solutions ofthe VM system. These solutions introduce the systems of nonlocal semilinear ellipticequations with boundary conditions. Applying the lower-upper solution method, weestablish the existence theorems for solutions of the semilinear nonlocal ellipticboundary value problem under corresponding restrictions on distribution function. Wealso give several examples of the solutions at the end of the chapter.

The bifurcation problem for the stationary solutions of VM system is considered inChapter 8. It is translated into the bifurcation problem of the semilinear elliptic systemand is studied as an operator equation in Banach space. Using a classical approach byLyapunov–Schmidt, the branching equation is derived and asymptotics of nontrivialbranches of solutions is studied. Here the principal idea is to study a potential BEq,since the system of elliptic equations is potential. Further investigation establishes theexistence theorem for the bifurcation points and reveals the asymptotic properties ofnontrivial branches of the solutions of VM system.

In Chapter 9, we discuss the general and linear Boltzmann equations and corres-pondence with hydrodynamics and quantum physics. Discrete Boltzmann models areinvestigated in Chapter 10, paying special attention to the models of interactionsbetween particles in relation with conservation laws and validity of H-theorem.

In Chapter 11, we study the Boltzmann equation’s symmetry. Here, commutationof collision operator with rotation group comes first, as it provides us with a solutionfor momentum system. An appendix to this chapter gives an example for nonlinearequations as convergent series for superposition of running waves.

Chapter 12 studies discrete models for gas mixtures with different particle massesand corresponding collision models. This applied problem is extremely important fornumerical modeling, as any appropriate model should be checked first to see if it iscompliant with conservation laws. Chapter 13 investigates the spectrum of Hamiltoni-ans in application to quantum optics. Here we can use the same “ideology” in applyingconservation laws that we used earlier, studying discrete models of Boltzmann equa-tion. This approach to the conservation laws reduces the dimension of the spectrumproblem to the finite-dimensional one. It has already been used by physicists in con-struction of frequency convertors.

Chapter 14 studies the stationary self-consistent problem of magnetic insulationunder space-charge limitation via the VM system. In a dimensionless form of the VMsystem, the ratio of the typical particle velocity at the cathode related to the velocityreached at the anode appears as a small parameter. The associated perturbation analysisprovides a mathematical framework to the results of Langmuir and Compton. Westudy the extension of this approach, based on the Child-Langmuir asymptotics tomagnetized flows.

Chapter 15 shows that when the VM system is written in Hamilton form usingnonlocal Poisson bracket, the use of Hamiltonian formal approach for the real kineticequations still is under discussion. This unique attempt — to approximate the Poissonbracket in VM system by a finite dimensional one — exists similarly between the VMsystem and the Liouville equation. Hence, the study of the approximate integrationmethods for analytically integrable and nonintegrable Liouville equations is a corner-stone for development of wavelet solutions for the VM system. Thus, we propose an

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Preface xiii

effective technique of approximate integration for the Cauchy problem of the generali-zed Liouville equation based on the orthogonal decomposition over Hermite polyno-mials and Hermite functions. The respective mean convergence theorems are proved.The importance of this approach is related to the possibility of automated analyticcomputations in modern mathematical packages such as Maple or Mathematica.

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About the Authors

Victor Vedenyapin

Vedenyapin graduated from Faculty of Mathematics andMechanics of Lomonosov Moscow State University in 1971(department of differential geometry). His Ph.D. thesis wasdefended in 1977 and D.Sci. Thesis in 1989. A list of pub-lications contains more than 100 titles. Since 1992, withM. Maslennicov and V. Dorodnitsyn, Vedenyapin has led theresearch seminar at the Keldysh Institute of Applied Mathe-matics on mathematical physics. Professor of Moscow Uni-versity of Physics and Technology since 1992. USSR StatePrize winner (1989) Mathematical Methods in BoltzmannEquation.

Fields of interest: kinetic equations; Boltzmann equation;Vlasov equation; entropy; Quantum Hamiltonians, Ergodictheory.

Alexandr Sinitsyn

Sinitsyn graduated from Irkutsk Polyteknical Institute on1983. His Ph.D Thesis was defended in 1989 and D.Sci Thesisin 2004. Professor of Colombian National University. He pub-lished 60 articles and 2 books. Co-director of INTAS researchproject “PDE modelling semiconductors.” Worked as visitingprofessor in Paul Sabatier University, Toulouse, France.

Fields of interest: kinetic, Boltzmann, Vlasov equationsand their applications.

Eugene Dulov

Dulov graduated from Faculty of Mathematics and Mechanicsof Lomonosov Moscow State University in 1993. His Ph.D.thesis was defended in 1997. He is a Lecturer of UlyanovskState University, Professor of Colombian National University.A list of publications contains 38 titles.

Fields of interest: numerical methods, kinetic, Vlasovequations and their applications, development of algorithms.

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1 Principal Concepts of KineticEquations

1.1 Introduction

Kinetic equations describe the evolution of distribution function F(t,v,x) of moleculesor other objects (like electrons, ions, stars, galaxy, or galactic aggregations) withrespect to their velocities v and space coordinates x at moment t. In particular, thismeans that a number of particles in the element of phase volume dvdx is given by aquantity F(t,v,x) dvdx. The simplest example equation known as an equation of freemotion is given below:

∂F

∂t+

(v,∂F

∂x

)= 0 (1.1.1)

and could be simply resolved—F(t,v,x)= F(0,v,x− vt).The goal of this book is to study the kinetic Boltzmann and Vlasov equations.

1.2 Kinetic Equations of Boltzmann Kind

The first kinetic equation to be studied was Boltzmann’s. It considers collision pro-cesses via collision integral added into (1.1.1):

∂F

∂t+

(v,∂F

∂x

)= J[F,F]. (1.2.1)

Collision integral J[F,F] is a quadratic operator, considering a pairwise collisionof particles. Equation (1.2.1) was obtained by Maxwell and Boltzmann for derivationof Maxwellian distribution by velocities. This result has been used for explanation ofClapeyron ideal gas law (see Section 1.4).

The already mentioned Maxwellian distribution is connected with one of the firstknown classic results for the Boltzmann equation (1.2.1)—a proof of the so-calledH- theorem. This theorem claims that functional

H[F]=∫

F lnFdvdx

Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00001-6c© 2011 Elsevier Inc. All rights reserved.

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2 Kinetic Boltzmann, Vlasov and Related Equations

does not increase for Boltzmann equation, i.e., dH/dt ≤ 0. Since H equals to entropywith negative sign, this fact was interpreted by Boltzmann as a proof that entropyincreases, i.e., a justification of the second thermodynamics law.

The inequality of H- theorem holds only sometimes, and we will discuss corre-sponding conditions later in Chapter 9. When entropy stays unchanged, we obtain aMaxwellian distribution. Hence, the H- theorem justifies not only a stationary stateof Maxwellian distribution, but tends to it also, its stability and the second thermo-dynamics law.

However, the Boltzmann equation was developed by Maxwell for more broad aims.The initial Maxwell goal was to obtain the equation of continuos medium (Navier-Stokes type) derived from the Boltzmann equation; namely, to define transport coef-ficients (heat and viscosity) and their dependence on intermolecular interaction. Hesucceeded with intermolecular interaction potential of the form U(r)= r−4 (the so-called Maxwellian molecules), when the collision integral becomes simple.

Boltzmann [52] and Hilbert (Hilbert method [67]) tried to state the similar resultsfor another potential, but never succeeded. However, Chempen and Enskog [67, 158]achieved this goal by means of special scheme based on perturbation theory, knowntoday as Chempen–Enskog method. The stakes were high, since a solution couldprovide qualitative forecasts in molecular-kinetic theory, an issue of hard criticismat that time. Their result predicted a thermal diffusion, and this issue was resolved.The tension was really high and the problem was discussed both by scientists (Machand Avenarius, for example) and philosophers and polititians. But the solution ofChempen and Enskog was a little bit late, because the Avogadro number was cal-culated by two different ways and the estimations were quite close. This justi-fied the molecular-kinetic theory at least for scientists and a scientific communitycalmed down.

At the present time, this equation with respective corollaries has several differentapplications, for example, modeling of the middle atmosphere layers. Tall atmospherelayers are well described by equation of free motion (1.1.1) (also called Knudsen orfree gas equation). Lower layers are described by gas dynamics equations, which alsoare derived from the Boltzmann equation. Derivation and numerical modeling of thetwo-layer models (see [158]) related with modeling of aircraft motion keeps themimportant. The other important application deals with chemical kinetics and mix-ture modeling especially, known as a discrete models of Boltzmann equations (seeChapters 10 and 12).

The other widely used corollary of Boltzmann equation is the transport equation,describing the scattering of particles on a fixed background. This is a linear Boltzmannequation. Such equations are used for description of neutrons transport in nuclear reac-tors and radiation transport in atmosphere when photons are scattered by medium.

The limit case of the Boltzmann equation, known as the Landau equation, appearswhen the main contribution in scattering cross-section is made by strong scatteringforward. It is used for plasma description.

There also are quantum analogs of the Boltzmann equation, called Uehling-Uhlenbeck equations. For these equations, Fermi-Dirak or Boze-Einshtain distribu-tions are steady state instead of Maxwell distribution. Therefore, one can represent thehierarchy of Boltzmann-type equations in the scheme seen in Figure 1.1.

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Principal Concepts of Kinetic Equations 3

Boltzmann equation

?

Discrete models Uehling-Uhlenbeck equation

Landau equation Transport equation

Figure 1.1 The hierarchy of Boltzmann type equations illustrates connections among them.

The lines marked with question signs outline the fact that corresponding equationsare still undiscovered (Landau approximation for Uehling-Uhlenbeck equations, forexample).

1.3 Vlasov’s Type Equations

Vlasov-type equations are compared with equations of the Boltzmann type describingshort-range interactions. The Vlasov equation or equation of self-consistent field hasthe form

∂F

∂t+

(v,∂F

∂x

)+

(f (F),

∂F

∂v

)= 0. (1.3.1)

Here the force f is the functional of a distribution function F, and equation (1.3.1)has the form of shift equation along characteristics. A simplest kind of functional(force f ) describing a dependence from distribution function corresponds to pairwiseinteraction potential K(x,y):

f =−∇∫

K(x,y)F(y,v, t)dvdy. (1.3.2)

This kind of interaction introduces the system of Vlasov equations. Gener-ally speaking, mostly we are using a phrase—“Vlasov plus something more”equations—aimed to distinguish between the kinds of interactions. There are Vlasov-Poisson, Vlasov-Maxwell, Vlasov-Einstein, and Vlasov-Yang-Mills equations (seeChapter 3 for details).

The Vlasov-Poisson equation exists for two kinds of problems: for gravitation andplasma. In both cases, we exchange a potential (1.3.2) by Poisson equation apply-ing a Laplace operator. Here K(x,y) assumed to be a fundamental solution [303]of a Laplace operator: 1xK(x,y)= δ(x− y). Therefore, K is a potential of a sin-gle charge in three-dimensional case (K(x,y)=− 1

4π |x− y|−1) (Coulomb law), of athread (K(x,y)= 1

2π ln|x− y|) for a two-dimensional, and plane (K(x,y)= 12 |x− y|)

in one-dimensional cases [303].When we study gravitational case exchanging Newton type interaction by the

Einstein’s one, we obtain a so-called Vlasov-Einstein equation.If we exchange electrostatic by electrodynamic interaction required for plasma, we

obtain Vlasov-Maxwell equations. If not a charge, but some vector characteristic is

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Vlasov-Poisson equation

Vlasov-Einstein equation Vlasov-Maxwell equation

Vlasov-Yang-Mills equation

Figure 1.2 The hierarchy of Vlasov equations.

unchanged (like isotopic charge or color), then we should take matrices instead ofelectromagnetic 4-potentials, thus, obtaining Yang-Mills equations. These equationsrepresent the modern theory of joint weak, electric and strong interactions.

Finally, we can introduce the following hierarchy of Vlasov equations:The given hierarchy (Figure 1.2) gives us an incredible example of interactions

between mathematics and several branches of natural sciences. Some of these con-nections will be presented in the following chapters, when we will study some basicsubstitutions into Vlasov equation:

l Equation of N bodies dynamics as a corollary of Vlasov equation obtained by substitutionof the sum of delta-functions;

l Substitution in the form of integrals of delta-functions and Lagrangian coordinates is usedfor oscillators and anti-oscillators and two-Hamiltonian structures;

l Euler-Lagrangian coordinates and hydrodynamic substitution is used in N-layer andcontinuum-layer hydrodynamics for modeling expanding universe, overlaping and evenapplicability of hydrodynamic description (see Chapter 2);

l Energetic substitution, when distribution function depends only from of energy (seeChapter 4). In this case, equation (1.3.1) is satisfied, and (1.3.2) transforms into nonlinearequation for potential. In applications, there were plasma diode (Lengmure diode), Debayequations for electrolytes and Len-Emden equation in gravitation. In math this equation ear-lier has been studied in geometry, and it is called Liouville equation. In two-dimensial caseit possesses a vast group of symmetries (conformal group).

The last energetic substitution gives equations similar to Bernoulli equations forEuler equations. Their fates are similar—they were initially discovered for particularcases. These equations then were studied before the generalized Vlasov equations hadbeen written. Moreover, they were introduced from the same energetic point of view,expressing the energy conservation law.

1.4 How did the Concept of Distribution Function ExplainMolecular-Kinetic and Gas Laws to Maxwell

Let F(t,v,x) be a distribution function of molecules by their velocities v ∈ R3 andspace x ∈ R3 at moment t. It means that quantity F(t,v,x)dvdx represent a number ofparticles in the element of phase space dvdx. Let us consider the following macro-scopic values:

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1. The density (a moment of zero order)

n(x, t)=∫R3

F(t,v,x)dv. (1.4.1)

2. The mean velocity (a first moment), as a mean value of velocity v:

u(x,y)=1

n

∫vF(t,v,x)dv.

3. Stress tensor (second centered moment):

pij = m∫(vi− ui)(vj− uj)F(t,v,x)dv, (1.4.2)

m—is a molecule mass.4. The quantity

∫vϕ(v)Fdv is called the flow of a value

∫ϕ(v)Fdv. For example, nu is a flow

of the density n.5. An auxiliary relation

∫vivjFdv= nuiuj+ pij/m connects velocity flows and stress tensor.

6. Kinetic energy of moving molecules is represented as a kinetic energy mu2/2 of moleculesmotion as whole entity plus kinetic energy of relative motion

E(x, t)=m

2n

∫v2Fdv=

mu2

2+

m

2n

∫(v− u)2Fdv.

The second term also may be interpreted as internal energy:

e=m

2n

∫(v− u)2Fdv. (1.4.3)

7. Pressure. The idea is to calculate which pressure makes a gas onto the unit ground witha normal n for a given distribution function F(t,v,x). We’ll assume that a ground witharea S reflects molecules like a mirror. Then the magnitude of changing molecular momen-tum with velocity v is 2m(v,n). The number of molecules impacting the ground during 1tequals F(v)dvV . Here V is a volume of a parallelepiped with a bottom S and a side v1t.Namely:

V = (v,n)1tS.

Hence, we define pressure p as

p=FORCE

AREA=

2m(v,n)S

×(v,n)SF1tdv

1t= 2m(v,n)2Fdv.

Integrating an obtained relation in v, we get the formula for pressure

p(n,x, t)=∫

(v,n)>0

(v,n)2Fdv. (1.4.4)

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Additionally, we would like to outline the relationship between the relation (1.4.4)and two famous physical laws.

Pascal law or independence of pressure from direction holds if p does not dependon n. The sufficient condition is isotropy of distribution function, i.e., when F dependsonly on v2.

Mendeleev-Klapeyron law or state equation:

p=2

3en. (1.4.5)

Comparing (1.4.3) and (1.4.2), we see that if one assumes p= (p11+ p22+ p33)/3,then (1.4.5) fulfills for any distribution function. Comparing (1.4.5) with formula(1.4.4) one deduces when (1.4.4) represent a real, physical pressure:

(a) The mean velocity u is zero. If not, we have to redefine (1.4.4) by exchanging v with v− u;(b) Isotropy of distribution function, when relation (1.4.4) does not depend on n.

1.5 On a Kinetic Approach to the Sixth Hilbert Problem(Axiomatization of Physics)

The Sixth Hilbert problem was formulated as a problem on development of axiomaticmethod in natural sciences. For example, it was partially solved by Kolmogorovfor the probability theory. But it hardly could be said to be as a well-studied prob-lem in physics. Better to be named nearly untouched. Nevertheless, its importanceis undoubted from the philosophy, natural sciences and even tutorial points of view.Any person assuming to derive one equation from the other gets involved with thisproblem.

For example, kinetic Boltzmann, Vlasov and Landau equations were derived byBogolyubov [49] using N–body dynamics (see also [67, 95, 227]). Another attemptwas made by Godunov [118], who proposed the hyperbolic point of view to classifythe fundamental equations found in the famous physical textbook volumes by Lifshitzand Landau. One can obtain hyperbolicity for the Euler-type equation derived fromBoltzmann equation as a first approximation in Chempen-Enskog method (see Chap-ter 9 and [116]). There the special twicely divergent form of Euler type equations isobtained from a simple discrete model of Boltzmann equation.

Following the mentioned ideas, we can propose as a basis of physico–mathematicaldescription of the world just two Lagrangians: Vlasov-Einstein and Vlasov-Yang-Mills (see Chapter 3 for details). The first one explains gravitational interactions, whilethe second one’s weak and strong electric interactions.

The dynamics of N bodies can be derived from them by an ordinary substitution,described in Chapter 2. That was already made before for modern introduction in clas-sical and statistical mechanics. Besides, all the main equations describing the aggre-gate states of a substance, like plasma, gas, fluid or rigid body also should be derived.Here plasma is described by Vlasov-Maxwell equation, which has been “split of” from

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the Vlasov-Yang-Mills equation. A diluted gas is decribed by Boltzmann equation,which could be obtained using the scheme by Bogolyubov [67, 95, 227] from thedynamics of N bodies. The equations of continuous medium (Euler and Navier-Stokes)are obtained from Boltzmann equation by Chempen-Enskog method.

In application to rigid bodies and fluids, quantum kinetic equations [177]are obtained from the correspondence “quantum Hamiltonians—kinetic equations”(see Chapter 13). This correspondence is not well developed yet (see [292] andChapter 13), but [177] uses it explicitely and implicitely. The quantum Hamiltoniansitself are obtained from the second Lagrangian by quantization of connected fields.

As a short summary, the above examples already became the cornerstones for thekinetic approach to a construction of modern physics. They were made by a synthesisof Landau and Lifshitz, Bogolyubov, and Godunov approaches.

The correspondence “quantum Hamiltonians—kinetic equations” already revealedthe possibility to generalize kinetic equations in applications to chemically reactingmixtures and problems with triple and higher number collisions (see Chapter 13). Thesame correspondence allowed also to obtain a simple formulas for conservation laws(Chapter 13) which led us to the method for constructing discrete models for mixtureswith correct number of invariants (see Chapter 12).

The other very important research instrument is a careful study of the special con-servation laws—linear in the number of particles. Precisely, such conservation lawsare the basics for:

l collision invariants of Boltzmann equation;l the uniqueness theorem of the Boltzmann H- function;l in conditions of chemical equilibrium;l in the studying of the sets of stationary positions of Markov processes and the Pauli Master

equation;l in the study of spectrums of quantum Hamiltonians;l in reasoning of the correspondence “quantum Hamiltonians—kinetic equations.”

1.6 Conclusions

1. The derivation of the Vlasov-type equations is not too complicated. Even for the mostadvanced Vlasov-Maxwell equation, one can use a more simple method, different fromBogolubov chains technology.

2. The derivation of the Boltzmann-type equations is complicated. So, we introduce only thecorrespondence principle allowing to solve the problems on interaction cuts (coefficients ofBoltzmann-type equations).

3. Axiomatization of physics (Sixth Hilbert problem) is very important from the philosophical,natural sciences and tutorial points of view as justification and classification of differentequations.

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2 Lagrangian Coordinates

2.1 The Problem of N-Bodies, Continuum of Bodies, andLagrangian Coordinates in Vlasov Equation

Let us consider Vlasov equation (1.3.1)–(1.3.2) when (1.3.2) is substituted into (1.3.1):

∂F

∂t+

(v,∂F

∂x

)−

(∇x

∫K(x,y)F(y,v, t)dvdy,

∂F

∂v

)= 0. (2.1.1)

Considering substitution

F(t,v,x)=N∑

i=1

ρiδ(v−Vi(t))δ(x−Xi(t)) (2.1.2)

for δ(x)—Dirac δ-function, Vi(t) and Xi(t) are time-dependent functions (coordinatesand velocities of particles), ρi > 0—numbers (weights of particles). When Vlasovstarted to study this equation he already knew that such substitution could be appliedif functions Xi and Vi satisfy the motion equations of N-bodies dynamics [306].

Xi =Vi

Vi =−

N∑j=1

∇1K(Xi,Xj)ρj,(2.1.3)

where ∇1 is gradient vector by the first argument.Consider a similar substitution, if the sum in (2.1.2) is changed by integral

F(t,v,x)=∫ρ(q)δ(v−V(q, t))δ(x−X(q, t))dq. (2.1.4)

Here, we have to define the right expression as generalized function. This could bedone naturally: this is a linear functional defined by the formula(∫

ρ(q)δ(v−V(q))δ(x−X(q))dq,ϕ(v,x)

)=

∫ϕ(V(q),X(q))ρ(q)dq.


http://dx.doi.org/10.1016/B978-0-12-387779-6.00002-8

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This is a quite wide generalization of a “simple layer” notion [303] for para-metrically defined surfaces (q→ (X(q, t),V(q, t)). Parameter q may run through anydomain in the space of arbitrary dimension Rk or through some kind of manifold. Usu-ally one takes k < 6. Let us establish conditions when the substitution (2.1.4) turns(2.1.1) into the equality.

Using relation vδ(v−V(q))= V(q)δ(v−V(q)) and replacing v by V(q, t) accord-ing to multiplication rule for δ function [303] we have in (2.1.1) for function (2.1.4):

∂F

∂t=

∫ρ(q)

(∇vδ(v−V(q)),−

∂V

∂t

)δ(x−X(q))dq+

+

∫ρ(q)δ(v−V(q))

(∇xδ(x−X(q)),−X(q)

)dq;(

v,∂F

∂x

)=

(v,∫ρ (q)δ (v−V (q))∇xδ (x−X (q))dq

)=

=

(V (q, t) ,

∫ρ (q)δ (v−V (q))∇xδ (x−X (q))dq

).

Finally,(∇x


∂F

∂v

)=

=

(∫∇1K(X(q, t),X(q′, t))ρ(q′),ρ(q)∇vδ(v−V(q))δ(x−X(q))dqdq′

).

Here ∇1 is a gradient over first argument. Here we used the definition of δ-functionwhile integrating over y. Integrating over v we used the property that an intergral ofδ-function is equal to one. Also, we changed x by X(q, t) using the multiplicationproperty of δ-function. Comparing all three expressions, we can represent them as

X(q, t)=V(q, t)

V(q, t)=−∫∇1K

(X(q, t),X(q′, t)

)ρ(q′)dq′.

(2.1.5)

Equations (2.1.5) naturally are called the equations of continuum body dynamics.Consider substitution (2.1.2) for N = 1. Then x= v, v=−ρ∇1K(x,x). If potential

K(x,y) is an even smooth function of the difference x− y, then ∇1K(x,x)≡ 0. In otherwords, it means that a body (particle, etc.) does not have an influence itself (nonself-acting); see [48] for additional details.

In the preface of Vlasov’s book [307], Bogolubov said that the Vlasov equation is acornestone of plasma physics, but it also has an exact solution like function (2.1.2), asin classical mechanics. Functions are exactly such a microscopic solution. Bogolubovalso proved that substitution (2.1.2) can be applied for the Boltzmann-Enskog equation[48], using dynamics of hard balls.

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Lagrangian Coordinates 11

Conclusions

1. The Vlasov equation accepts the substitution of the form (2.1.2) and contains within itself adescription of the motion of N bodies for arbitrary number N. Thus, it proves the equationto be fundamental.

2. The Vlasov equation accepts the substitution of the form (2.1.4) providing equations forcontinuum of bodies. If q= (X(0),V(0)) the initial coordinates, then q is called Lagrangecoordinates and equation (2.1.5) is interpreted as a transformation of it.

2.2 When the Equations for Continuum of Bodies BecomeHamiltonian?

A useful candidate to be a Hamiltonian for equation (2.1.5) is a functional of theform

H[P,X]=∫

P2(q)

2ρ(q)dq+

1

2

∫K(X(q),X(q′)

)ρ(q)ρ(q′)dqdq′. (2.2.1)

This functional obviously is a generalization of the Hamiltonian of N bodies. Herewe exchange the sum by integral. Let us find, when equations (2.1.5) are Hamiltonian,i.e., they comply with the system

X(q, t)=δH

δP(q)

P(q, t)=−δH

δX(q).

(2.2.2)

On the right side of the expression, we use the variational derivative, analogous to thepartial derivative:

Let F[g] be some functional. Then its weak differential is

F′[g]h=d

dλF(g+ λh)

∣∣∣∣λ=0

.

We call a variation derivativeδF

δg(x)of the functional F as a function, defined by

the following equality:

F′[g]h=∫

δF

δg(x)h(x)dx.

This equality can be interpreted in the sense of generalized functions [303]. In thiscase, we will call variational derivative—a linear functional acting according to theformula(

δF

δg(x),h(x)

)=

d

dλF(g+ λh)

∣∣∣∣λ=0

.

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Hence, equation (2.2.2) also should be considered in the sense of generalized functions(i.e., the equility of the functionals).

Exercise 2.1. Derive (2.2.2) from (2.2.1).

Solution.δH

δP(q)=

P(q)

ρ(q);

δH

δX(q)=−

1

2

∫∇1K

(X(q),X(q′)

)ρ(q)ρ(q′)dq′−

−1

2

∫∇2K

(X(q′),X(q)

)ρ(q)ρ(q′)dq′. (2.2.3)

We can see, that this equation coincides with (2.1.5) if

(a) V(q)=P(q)

ρ(q);

(b) Integrals in expression (2.2.3) are equal.

The first condition represents the natural condition that velocity V and impulse Pare related; the second one is fulfilled if potential K is symmetric with respect to itsarguments K(x,y)= K(y,x).

Conclusion. If potential K is symmetric, then the equations for the continuum of bod-ies (2.1.5) are Hamiltonian with a multiparticle Hamiltonian, defined by (2.2.1). Theylead us to the Vlasov equation (2.1.5) in Lagrangean coordinates (2.2.2).

Another characteristic Hamiltonian is given by expression

H =v2

2+U(x, t), where U =


and defines Vlasov equation (2.1.1) of the form∂F

∂t+{H, f } = 0 in Euler coordinates.

Here {H, f } is a Poisson bracket {H, f } =∂H

∂v

∂f

∂x−∂f

∂v

∂H

∂x.

2.3 Oscillatory Potential Example

Let us investigate a special case in which K(x,y)=ω2

2(x− y)2.

The problem has an exact solution in Lagrange coordinates. Equations (2.2.2) aresimplified to

X(q, t)+ω2ρ0X(q, t)= ω2∫

X(q′, t)ρ(q′)dq′.

Here, ρ0 =∫ρ(q)dq.

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X

(a) (b)

X

VV

Figure 2.1 Single dimensional (a) elliptic and (b) hyperbolic phase portraits.

Multiplying by ρ(q) and integrating over dq, we obtain an equation for middlecoordinate Q= 0, where Q(t)=

∫X(q, t)ρ(q, t)dq.

Then

X(q)= A(q, t)cos(ω√ρ0t)+B(q)sin(ω

√ρ0t)+

Q(0)

ρ0t+

Q(0)

ρ0.

So, one can see that particle system with oscillatory potential oscillate around com-

mon center with ω√ρ0 frequency. This center, in its turn, moves with a velocity Q(0)

ρ0.

Coordinate q enumerates particles and defines an amplitude.Assume Q(0)= Q(0)= 0. Drawing the phase portrait picture for single dimen-

sion, we will see that all particles have an elliptic trayectory, moving clockwise; seeFigure 2.1a.

The rotation formula in

(X

V/(ω√ρ0)

)is given below:

(X(q, t)

V(q, t)/ω√ρ0

)=

(cos(χ t) sin(χ t)−sin(χ t) cos(χ t)

)(X(q,0)

V(q,0)/ω√ρ0

).

Here, χ = ω√ρ0.

2.4 Antioscillatory Potential Example

The antioscillatory potential is the same the thing as before, just taking the differentsign:

K(x,y)=−ω2

2(x− y)2.

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Obviously, the solution looks similar if we exchange trigonometric functions withhyperbolic ones:(

X(q, t)V(q, t)/ω

√ρ0

)=

(ch(χ t) sh(χ t)−sh(χ t) ch(χ t)

)(X(q,0)

V(q,0)/ω√ρ0

).

Here, χ = ω√ρ0.

The corresponding phase portrait (see Figure 2.1b) represents the divergent hyper-bolic movement with asimptotes X =±V/ω

√ρ0.

Later we’ll study this solution again, comparing this trajectory with real ones thatappear in plasma and gravitational modeling problems.

2.5 Hydrodynamical Substitution: Multiflow Hydrodynamicsand Euler-Lagrange Description

We call as hydrodynamical substitution the substitution introduced in [259]

F(t,v,x)= ρ(x, t)δ (v−V(x, t)) .

This analytical expression means one simple thing: at each point x there exists only onevelocity value V(x, t). In particular (singleflow hydrodynamics), it gives the system ofequations for ρ and V:

∂ρ

∂t+ div(Vρ(x))= 0,

∂V

∂t+Vi

∂V

∂xi+

∫∇xK(x,y)ρ(y, t)dy= 0.

(2.5.1)

The mentioned substitution could be generalized. Namely, N-flow hydrodynamicsis defined when [259]

F(t,v,x)=N∑

l=1

ρl(x, t)δ (v−Vl(x, t)) .

Also, we can regard continuum-layered, or continuum-flow hydrodynamics, when thesum is replaced by integral

F(t,v,x)=∫ρ(x,s, t)δ (v−V(x,s, t))ds.

The right part of the expression denotes generalized function providing the value∫ρ(x,s)ϕ (x,V(x,s))dsdv

for the testing function ϕ(x,v); s—Lagrange coordinate.

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Now we can derive the equations over ρ and V . First, we need to evaluate all termsin (2.1.1):

∂F

∂t=

∫∂ρ

∂tδ(v−V)ds+

∫ρ

(∇vδ(v−V),−

∂V

∂t

)ds,(

v,∂F

∂x

)=

(v,∫(∇xρ)δ(v−V)ds

)+

+

(v,∫ρ∇iδ(v−V)

(−∂Vi

∂x

)ds

)(α)=

=

(v,∫(∇xρ)δ(v−V)ds

)+

∫ (V,∫ρ∇iδv(v−V)

(−∂Vi

∂x

))ds−

−

∫δij∂Vi

∂xjδ (v−V)ρ(s)ds,(∫

∇x


∂F

∂v

)=

=

(∫∇xK(x,y)ρ(y,s)dyds,

∫ρ(x,q)∇vFdq

).

In the second equility (α), we changed v by V using the multiplication rule for gradientof the δ-function

vj∇iδ(v−V)= Vj∇iδ(v−V)− δijδ(v−V).

Collecting δ-function members, we obtain the system of equations of variableρ(x,q, t)

∂ρ

∂t+ (V,∇xρ)+ ρ divV = 0. (2.5.2)

The second equation of variable V is obtained by collection of δ-function gradientcoefficients:

∂V

∂t+Vi

∂V

∂xi+

∫∇xK(x,y)ρ(y,q, t)dydq. (2.5.3)

Equations (2.5.2) and (2.5.3) are the equations of continuum-layered hydrodynamics.One should note that this equation is not valid, due to ambiguity of the functions

when the number of layers differs from point to point and is time-dependent (so-calledoverlapping). The last case corresponds to a free motion, when K = 0. Nevertheless,we can use a Lagrange descriptive technique to investigate it.

Example 2.1. Free motion. Assuming K = 0 in (2.1.1), we get (1.1.1).

A. There is no overlapping for any moment of time (Figure 2.2a).B. The overlapping occurs when fast-moving particles leave the slower ones behind

(Figure 2.2b).

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X

V

X

V

(a) (b)

Figure 2.2 (a) Non overlapping and (b) Overlapping cases.

Equations (2.5.1) and (2.5.2), (2.5.3) cannot describe the situation after the over-lapping, but the initial ones (2.1.1), (2.1.5) still are good.

Exercise 2.2. Find overlapping moments using different types of coordinate X(s,0)and velocity V(s,0) dependencies in Lagrange coordinates s.

Example 2.2. Let F(x,v,0)= ρ(x,0)δ(v−V(x,0)) at the initial moment is definedby conditions

ρ(x,0)=

{1 |x| ≤ 1,0 |x|> 1,

V(x,0)= x2 a branch of parabola.

Using Lagrange coordinates, we get X(s,0)= s, V(s,0)= s2 and the movement equa-tion becomes X(s, t)= 0. It can be solved explicitly: V(s, t)= s2, X(s, t)= s2t+ s. Tofind the overlapping moment, we have to study when ∂X

∂V turns to zero for the first timeor when the gradient catastrophe occurs when ∂X

∂V turns to infinity. Reducing s, weget X(V, t)= Vt−

√V for the left branch. Differentiating, we obtain ∂X

∂V = t− 12√

V.

Hence, function ∂X∂V turns to zero for the first time on the segment 0≤ V ≤ 1 for V = 1,

t = 12 . After the moment t = 1

2 function V(x, t) is not uniquely defined, and the hydro-dynamic description is of no use while the solution in Lagrange coordinates is kept.

Overlappings also are the basics for the disk theory of a large-scale universe. Mak-ing a projection on the x coordinate, one gets a density singularity, approaching thebasics of the Lagrange singularities theory [15].

Exercise 2.3. Derive from equation (2.1.1) a hydrodynamic-type equation assumingf (x,v) to be an arbitrary force instead of a self-consistent field.

2.6 Expanding Universe Paradigm

The term expanding universe first appeared after the Friedman’s solutions of Einsteinequations and Hubble’s discovery of red shift. There exists a simple nonrelativistic

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analogous solution, known as a “self–gravitating” ball, known also as the Miln-McKree model [319]. In addition this is an exact solutions of Vlasov-Poisson equationin gravitational field of the form

∂F

∂t+

(v,∂F

∂x

)−

(∇u,

∂F

∂v

)= 0,

1u= 4πγ∫

F(x,v, t)dv.

Here the quantity γ is a well-known gravitational constant.Here solutions can be written as

F(t,v,x)=∫δ(v−V(q, t))δ(x−X(q, t))dq

while X(q, t) satisfies the equation

X(q, t)=−γ∫

dq′

|X(q)−X (q′)|.

The considered solutions are spherically symmetric, since X(q, t)= q|q|R(r, t), r =|q|.

The particle velocities are directed along radius vectors and depend only on radius.Hence, the movement equation takes the form

R(r, t)=−γM(r)

R2(r, t).

Here M(r) is a mass of the ball with a Lagrange coordinate less than r. Thus, r couldbe interpreted as a spherical layers enumerator.

Here we used the known fact, that a uniform spherical layer applies zero force toany point inside it. The force of attraction of outside points exactly equals to the forceof attraction by the same mass put in center of the layer. Integrating this equation, weobtain

1

2

(R)2− γ

M(r)

R(r, t)= C. (2.6.1)

If we take C > 0 (see Figure 2.3), then we have an unbounded expansion, alsocalled an open model. Assuming C < 0, an expansion is exchanged by contraction,providing us with closed or oscillating Universe model (compare with Sections 2.3and 2.4).

Dividing equation (2.6.1) by R2, we get(R

R

)2

= H2(t)= 2γM(r)

R3(r, t)+

2C

R2,

where quantity H(t) is called Hubble constant. One should know that definition of thisquantity was made initially in 1929 assuming the expansion rate of Universe to beconstant. Advanced models are time-dependent, but they also mean to use the samehistorical name Hubble constant.

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X

V

C > 0

C < 0

U(R) = −γ MR

Figure 2.3 Open and closed models of the Universe.

Ratio on the left is measurable but implies a high error level (the most recent andreliable estimation of the constant was made in 2003 by WMAP satellite). Regardingthe expression on the right, it is proportional to the density ρ = 3M

4πR3 . Taking C = 0,

we get critical density ρ = H2

2γ3

4π . Higher densities lead to the closed models, lowerones—to the open model. The most important fact behind this is the density ρ.It’s assumed to be independent of r [319], and the uniform model of universe in otherwords. As a good practice, we recommend a reader spend some time thinking aboutvariable universe density and effects for physics and cosmology.

2.7 Conclusions

This chapter is devoted to the study of the construction scheme for the theory ofVlasov’s kinetic equations. We studied in detail the connections of Vlasov-Poissonequations with hydrodynamic equations and equations with Hamiltonian dynamics,thus, justifying the basis of particle method. A detailed study of Vlasov-Maxwell equa-tion is in Chapter 3.

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3 Vlasov-Maxwell and Vlasov-EinsteinEquations

3.1 Introduction

The present chapter is devoted to derivation and justification of the Vlasov-Maxwellsystem of equations. Initially developed by Vlasov in [304, 305], today it is widelyused for plasma description and modeling. The justification of the Vlasov-Einsteinequation is similar, and we touch on this briefly.

An interested reader should take into account that different authors introduce underthe name of Vlasov-Maxwell different equations, such as equations with relativisticor nonrelativistic dependence of momentum on velocity (see [18, 157]), for example.In [177] the dependence of velocity from impulse is not defined.

Therefore, it is important to establish a connection of this equation with classicalLagrangians in order to define the equation accurately and interpret the nature of theinvolved approximations. We will proceed with this in Section 3.5, presenting theshortest way (perhaps), revealing the connection of Vlasov-Maxwell equations andLagrangian ones in electromagnetism, introduced in [169].

Sections 3.2–3.4 were between meant to be auxiliary, since the process of deriva-tion in Section 3.5 is not uniquely defined. In Section 3.2, we obtain the equationsfor distribution function of particles shifting along trajectories of arbitrary dynamicssystem xi = Xi(x).

In Section 3.3, the Euler-Lagrange equation is studied: we assume an action tobe a path, and we justify the choice of distribution function in variables x,p (space–impulse).

In Section 3.4, the form of invariant measures in variables x,v (space-velocity) isexplained.

3.2 A Shift of Density Along the Trajectoriesof Dynamical System

Let us consider an arbitrary dynamical system, i.e., a system of nonlinear differentialequations in k-dimensional space:

xi = Xi(x), i= 1, . . . ,k. (3.2.1)

Kinetic Boltzmann, Vlasov and Related Equations. DOI: 10.1016/B978-0-12-387779-6.00003-Xc© 2011 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/B978-0-12-387779-6.00003-X

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Assume that we allocated particles according to some initial density function f (0,x).This means that at moment t this density becomes f (t,x), and the number of particlesinside the domain G is

N(G, t)=∫G

f (t,x)dx.

Hence, the main question is: How does f (t,x) evolve?We want to prove that corresponding equation becomes

∂f

∂t+

∂

∂xi( fXi)= 0. (3.2.2)

It is assumed that summation should be done with respect to both upper and lowerindexes.

3.2.1 Method 1. Dirac’s δ-Function Method

Consider the distribution function of N particles, being shifted by the trajectories ofthe system

f (t,x)=N∑

l=1

δ (x− xl(t)),

where for each fixed l function xl(t) satisfies equations (3.2.1). Then, differentiating itin time, we obtain

∂f

∂t=

∑(∇xδ(x− xl), (−xl)).

By contrast, using the following expression to evaluate the divergence

d

dx(ρ(x)δ(x− x0))= ρ(x0)δ

′(x− x0),

we have

div( fX)=N∑

l=1

(∇xδ(x− xl),X(xl)).

Adding obtained expressions, we see that equations (3.2.2) are satisfied.Equality (3.2.2) also holds for an arbitrary function f (in a weak sense) when one

calculates the limit for the respective approximation as the sum of δ-functions.

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Vlasov-Maxwell and Vlasov-Einstein Equations 21

3.2.2 Method 2. Balance of Particles

The increase of particles velocity in domain G is

∂N(G, t)∂t

=−

∫∂G

(f EX,n

)ds. (3.2.3)

As it follows from (3.2.3), on a small segment of the boundary ds, the quantity ofoutgoing particles at time dt is fdsdt

(EX,n

), because all outgoing particles sweep out

the cylinder with bottom ds and side EXdt. Hence, its height is(EX,n

)dt. A minus sign

takes place in (3.2.3) since a normal vector is outward, and we calculate outgoingparticles while the left part of (3.2.3) counts the number of particles contained indomain G. Exchanging in (3.2.3) integral over surface by volume integral (using aStokes formula), we obtain the equation (3.2.2) integrated over domain G. Taking intoaccount that domain G was arbitrarily chosen, equation (3.2.2) is feasible.

If we rewrite equation (3.2.2) in the form

∂f

∂t+

(X,∂f

∂x

)+ f divX = 0 (3.2.4)

and divX = 0, then the left part of (3.2.4) is the total derivative of f (t,x) in time.

3.2.3 Conclusion

An equation describing the distribution function of the particles shifting along trajec-tory of dynamic system (3.2.3) has the form (3.2.4).

3.3 Geodesic Equations and Evolution of DistributionFunction on Riemannian Manifold

Let us consider the metrics gijdxidx j in the space Rn, x ∈ Rn, gij(x)—n2 functions.This means that the length of curve is defined by the formulas ([98, 169]):∫ √

gijxix jdt, (3.3.1)

and the geodesic equation is obtained from the principle of least action (the principle ofleast length). Generally speaking, action is written in the form S=

∫L(x, x)dt, where

L is Lagrangian. Then after the Euler-Lagrange equations are given by variation withfixed endpoints of trajectories (see [98], for example),

δS=∫δLdt =

∫ (∂L

∂xδx+

∂L

∂ xδx

)dt =

∫ (∂L

∂x−

d

dt

∂L

∂ x

)δxdt.

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We then obtain Euler-Lagrange equations:

d

dt

(∂L

∂ xk

)=∂L

∂xkk = 1,2, . . . ,n.

In the case of geodesic equation L=√

gijxix j, hence,

d

dt

(gkix j√gijxix j

)=

1

2√

gijxix j

∂gij

∂xkxix j. (3.3.2)

The functional of the length is invariant with respect to change t = ψ(τ) for anysmooth function ψ(τ), and the same property has equations (3.3.2).

Sometimes this property is used to simplify the equations analytically as far aspossible. We choose [98, 169] the length of line (interval, own time) s as a parame-ter τ obtaining ds=

√gijdxidx j. Hence, dividing by ds, one obtains

√gijxix j = 1 and

equations (3.3.2) are reduced to

d

ds

(gkix

i)=

1

2

∂gij

∂xkxix j. (3.3.3)

The last ones coincide with a Euler-Lagrange equation for action defined byLagrangian L= 1

2 gijxix j.Further transformation gives

xk=−0k

lmxlxm, 0klm =

gki

2

(∂gil

∂xm+∂gmi

∂xl−∂glm

∂xi

). (3.3.4)

Here gkl is a matrix, inverse to gij, and values 0klm are called Christoffel symbols.

Now we can write down equations (3.2.2) for distribution function f (x,v,s) overspace and velocities with length s, instead of time by analogy with (3.3.4):

∂f

∂s+ vi ∂f

∂xi−0l

ijviv j ∂f

∂vl−

∂

∂vl

(0l

ijviv j)

f = 0. (3.3.5)

The last term on the left side satisfies the fact that system (3.3.4) possesses nonzerodivergence. The transformation to the divergence-free representation could be done intwo ways.

3.3.1 Method 1. Coordinate–Impulse Change of Variablesand Hamiltonian Formalism

We introduce the impulses in an ordinary way [98]. If L= gijxix j

2 (this Lagrangiangives the same motion equations as (3.3.1)), then impulses are defined as pi =

∂L∂ xi =

gijx j and Hamiltonian H = pivi−L= pipjgij

2 .

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Then equations (3.3.3) become Hamiltonian:dx j

ds=gjlpl

dpk

ds=−

1

2

∂gij

∂xkpipj.

Exercise. Show that for any Hamiltonian system the divergence of the system isequal to zero.

Solution.xi=∂H

∂pi

pi =−∂H

∂xi

⇒ divF =∂2H

∂xi∂pi−

∂2H

∂pi∂xi= 0.

So, we obtain the equations (3.2.2) for distribution function f (s,x,p) in the coordi-nate and impulse space of the form:

∂f

∂s+ gkrpr

∂f

∂xk−

1

2

∂gir

∂xkpipr

∂f

∂pk= 0. (3.3.6)

This equation also reads∂f

∂s+{H, f } = 0, where {H, f } is a Poisson bracket:

{H, f } =∂H

∂pi

∂f

∂xi−∂f

∂pi

∂H

∂xi.

3.3.2 Method 2. Invariant Measure in Coordinate-Velocity Space

Let g be determinant of matrix gij. We introduce a new distribution function

F(x,v,s)=f (x,v,s)

g,

instead of f in equation (3.3.5).

Exercise 3.1. Show that for new distribution function the evolution equation isdivergence-free, and it has the form

∂F

∂s+ vi ∂F

∂xi−0l

ijviv j ∂F

∂vl= 0.

Solution. Using the differentiation rule of the determinant, the second term in(3.3.5) will be transformed as follows:

vk ∂g

∂xk= vk ∂gil

∂xkgikg

(α)= 20l

mlvmg.

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While equating (α), we applied an identity

0lml =

1

2gli(∂gil

∂xm+∂gmi

∂xl−∂glm

∂xi

)=

1

2gli ∂gli

∂xm.

For the new distribution function, the number of particles is written as

N(G, t)=∫G

F(x,v, t)g(x)dxdv.

Therefore, gdxdv is an invariant measure since F do not increase, i.e., its total deriva-tive is zero, and then the number of particles is conserved. A measure gdxdv is alsoconserved.

3.3.3 Conclusion

One can take impulses or velocities as the variables in distribution function and time orinterval s as time variable. In Section 3.3, for simplicity of equations we took an inter-val called a characteristic time [169] in relativity theory. The possibility to select s as aparameter means the synchronization of characteristic time for different particles, alsoknown as “twins paradox.” The one with a smaller characteristic time interval, or theone who was moving more will be found younger (this follows from the formula ds

cdt =

dsdx0 =

√1− v2

c2 for Mincowski metrics (1,−1,−1,−1), see [169] for details). We seethat minimal action principle with action (3.3.1) is identical to minimal time Fermatprinciple where time is interval or own time. Therefore, choosing s as a variable isformally allowed, but it does make the interpretation of results a bit complicated.

Exercise 3.2. Write down the equations for free particles distribution function depend-ing on interval and time as arguments. Compare them.

3.4 How does the Riemannian Space Measure BehaveWhile Being Transformed?

Let us develop change of variables xk= f k(ξ). The metric transformation becomes

(see [98] for example)

gijdxidx j= gij

∂xi

∂ξ l

∂x j

∂ξ rdξ ldξ r

= glr(ξ)dξldξ r.

Thus, g= J2g, where J is det(∂xi

∂ξ r

). As it follows, taking dx= |J|dξ , we get

√|g|dx=√

|g|dξ .Thenafter

√|g|dx is transformation invariant. Differentiating by parameter, one

obtains xk=

∂xk

∂ξ l ξl. Hence dV = |J|dv and gdxdv=

√|g|dx√|g|dv is invariant mea-

sure with each of the factors to be invariant with transformations.

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Conclusion. It is convenient to take impulses as the variables of distribution func-tion. In Section 3.5 we take time as a parameter τ . So time t, space coordinate xand impulses p will be taken as variables for distribution function f = f (t,x,p) inSection 3.5.

3.5 Derivation of the Vlasov-Maxwell Equation

The system of Vlasov-Maxwell equations describes motion of particles in their ownelectromagnetic field. We start from ordinary action inside magnetic field (see Section27 in [169]), Vlasov-Maxwell or Lorentz action. Here summation goes over repeatingupper and lower indexes:

SL = SVM = −∑α

mαc∑

q

T∫0

√gµν Xµα (q, t)Xνα(q, t)dt+

+

∑α

eαc

∑q

T∫0

Aµ (Xα(q, t)) Xµα (q, t)dt+

+1

16πc

∫FµνFµνd4x= Sp+ Sp−f + Sf , (3.5.1)

where Sp denotes the action of particles, Sf —an action of the fields, Sp−f particles-fields action.

Here α denotes a kind of particles, differing in mass mα and charge eα , q enu-merates particles inside of the kind, Xµα (q, t), µ= 0,1,2,3, q= 1, . . . ,Nα , α = 1, . . . ,r—4 coordinates of q-th particle of the kind α. Aµ(x) - potential, Fµν = ∂νAµ−∂µAν—electromagnetic fields, gµν—Minkowski metrics: gµν = diag(1,−1,−1,−1),i.e., diagonal matrix with 1 on the fist place and −1 on the others.

Variation will be calculated by special method [169]. First, we obtain the motion ofa particle in the field, afterwards, the motion of field with given motions of particles.Afterwards, we proceed with distribution functions for particles to obtain the requiredsystem of equations.

Step 1

The variation of Sp+ Sp−f in coordinates Xµα (q, t) give the motion equations of charges

in the field. We rewrite√

gµν XµXν for Minkowski metrics. Here the Greek indices runfour values µ,ν = 0,1,2,3; the Latin ones i, j—three: i= 1,2,3:

Sp =−∑α

mαc2∑

q

∫ √1−

x2α

c2(q, t)dt =

∫Lpdt,

where Lp is the Lagrangian of the particles.

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Here

x2= v2= x2

1+ x22+ x2

3 =−(x1x1+ x2x2

+ x3x3)=−xixi, i= 1,2,3,

a three-dimensional square of velocity; we keep in mind that x0= ct and we kept c2

outside the root. Varying this expression (we omit α here) leads us to

δSp = mc2∑

q

1

c2

∫xiδxi√

1− v2/c2dt = m

∑∫d

dt

(xi√

1− v2/c2

)δxidt.

Varying Sp−f (and omitting α again):

Sp−f =e

c

∑q

δ

∫ [cA0(x(q, t))+Ai(x(q, t)x

i(q, t)]

dt =

=e

c

∑q

∫ [c∂A0

∂xiδxi+∂Ai

∂x jxiδx j−

(d

dtAi

)δxi]

dt.

Hence, applying a condition δ(Sp+ Sp−f )= 0, we obtain the motion equation ofcharged particle in the field:

dpαi

dt=−

eαc

∂Ai

∂t−∂A0

∂xieα −

eαc

Fijxjα,

where

pαi =∂Lp

∂xiα

=mα xαi√

1− x2α/c

2, Fij =

∂Ai

∂x j−∂Aj

∂xi.

Step 2

The equation on distribution function is obtained by making the shifts of equationalong trajectories of the just defined dynamic system of motion on the particles inthe field. It is convenient to take distribution function from impulses, and not fromvelocities. First we need the expressions, defining the velocities via impulses:

pi =mvi√

1− v2/c2⇒ p2

=m2v2

1− v2/c2.

Denoting 1− v2/c2= γ−2, we obtain γ 2

= 1+ p2/(m2c2) and vi = pi/(γm).Hence, we found the equation for distribution function fα(x,p, t) analogous to (3.2.4).

∂fα∂t+

(vα,

∂fα∂x

)+

(−eα

∂A0

∂xi−

eαc

Fijvjα

)∂fα∂pi= 0. (3.5.2)

Here vαj =pαjmα

1√1+p2/(m2

αc2). We used the identity divp(Fi

jvj)= 0 also.

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This equation in [157, 177, 304] is written for ions and electrons in the followingform:

∂fe∂t+

(v,∂fe∂x

)− e

((E+

1

c[v,B]

),∂fe∂p

)= 0,

∂fi∂t+

(v,∂fi∂x

)+ ze

((E+

1

c[v,B]

),∂fi∂p

)= 0.

(3.5.3)

Here fi(t,p,x)—ion distribution function over the space and impulse coordinates atmoment t. Please note, that subindex i in (3.5.3) means the first letter of the wordion, not the usual dimension or summation index. fe(t,p,x)—electron distributionfunction; +ze the ion charge; (−e)—electron charge. [v,B]—the vector product.Books [157, 177, 304] do not define expression v via p, but it is usually taken as aclassical one vαj = pj/mα (see Chapter 4 or book [18], for example). Then it is conve-nient to write equations for distribution function f (t,v,x) in terms of velocities insteadimpulses. Velocities v in (3.5.3) has to be taken differently dependent upon impulsesfor electrons and ions. So we have to write ve = ve(p) and vi = vi(p) instead of v inequations (3.5.3) correspondingly.

Step 3

Equation for fields. In general, we follow the book [169], but we will use the distri-bution function instead of density. At first, we need to rewrite Sp−f via distributionfunction, making the sequence of transformations∑

q

→

∫dq→

∫f (x,p)dxdp,

delivering Sp−f written in the form

Sp−f =∑ eα

c2

∫Aµ(x)v

µα fα(x,p)d

3pd4x.

Variating by potentials Aµ(x):

δSp−f =∑ eα

c2

∫δAµ(x)v

µα fα(x,p)d

3pd4x,

δSf =1

16πc2

∫δFµνFµνd4x=

1

8πc2

∫δAµ∂νFµνd4x.

We assume δ(Sp−f + Sf )= 0 and obtain then

∂µFµν =−4π

c

∑α

eα

∫vµα fα(x,p)d

3p. (3.5.4)

The system of equations (3.5.2), (3.5.4) is known as Vlasov-Maxwell system.

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Remark 3.1. Equation (3.5.4) is the second from the couple of Maxwell equations.The first one follows from equalities Fµν = ∂νAµ− ∂µAν . Using the equivalent formin antisymmetric tensors differentiation notations [98], it is written as Fµνdxν ∧ dxν =2d(Aµdxµ). Hence, the first equation of the Maxwell couple looks like d(Fµνdxµ ∧dxν)= 0.

Remark 3.2. While deriving Vlasov-Maxwell equations using Bogoluybov’sscheme [47], we have to start with Hamiltonian systems with Lienart-Vihert potentials,known as retarded potentials. The corresponding Lagrangian for the weak relativity iscalled Darvin Lagrangian [227].

Remark 3.3. One can obtain Vlasov-Yang-Mills equations in a similar way byexchanging numbers with matrices instead of four potentials Aµ.

Conclusion

The system of Vlasov-Maxwell equations is obtained by variation of electromag-netic action (Lorentz action) with transition to distribution function. Equation fordistribution function is obtained by shifting the equation along particles motiontrajectories.

3.6 Derivation Scheme of Vlasov-Einstein Equation

Let us consider the action for the particle in gravitation field and for field [169]:

SVE =

∫ √gµν XµXνdqdt +

∫√−gRd4x= Sp+ Sf. (3.6.1)

Here R is a curvature [169]; variation by metrics is made via reperesenting the firstterm in Eulerian coordinates

SVE =

∫ √gµνvµvνF(p,x)d3pd4x +

∫√−gRd4x. (3.6.2)

Thus, as in the previous section, we obtain equation for field. Varying the trajec-tories of particles for Sp in (3.6.1), one obtains the equation for its motion in gravita-tional field. The equation for distribution function is just an equation for shifts alongthe characteristics.

3.7 Conclusion

The Vlasov-Maxwell and Vlasov-Einstein equations are obtained by uniform varia-tional method for the corresponding Lagrangians of electromagnetic and gravitationalfields.

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4 Energetic Substitution

4.1 System of Vlasov-Poisson Equations for Plasmaand Electrons

Let us consider the system of Vlasov-Maxwell equations over potentials Aν . Assum-ing ∂νAν = 0, known as Lorentz calibration, we obtain the wave form of relativisticVlasov-Maxwell system of [169]:

∂fα∂t+

(vα,

∂fα∂x

)+

(−eα

∂A0

∂xi+

eαc

Fijvjα

)∂fα∂pi= 0,

α = 1, . . . ,n, fα = fα(t,x,p), x,p ∈ R3,

1

c2

∂2Aν∂t2−1Aν =

n∑1

4πeαc

∫vαfαd3p.

(4.1.1)

Here Maxwell equations are reduced according to [169], and

vα =p

mα

(1+

p2

m2αc2

)− 12

.

For nonrelativistic limits, we handle vα = p/mα . The distribution function f (t,x,v)is usually regarded over velocities, providing us with the system of Vlasov-Poissonequations:

∂fα∂t+

(v,∂fα∂x

)−

eαmα

(∂U

∂x,∂fα∂v

)= 0,

α = 1, . . . ,n, fα = fα(t,x,v), (x,v) ∈ R3×R3,

1U =−n∑1

4πeα

∫fα(v,x, t)dv.

(4.1.2)

Quantity n= 1 is taken for the electron problem, and n= 2 for plasma, consisting ofboth ions and electrons.


http://dx.doi.org/10.1016/B978-0-12-387779-6.00004-1

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4.2 Energetic Substitution and Bernoulli Integral

Let the distribution function fα is a function of energy:

fα(v,x)= gα(mα

2v2+ eαU

).

Here gα is an arbitrary nonnegative function of energy. Then the first equation (4.1.2)is satisfied, and we obtain nonlinear elliptic equation over potential U(x):

1U = ψ(U), (4.2.1)

where

ψ(S)=−4πn∑1

eα

∫gα

(mαv2

2+ eαS

)dv. (4.2.2)

The given substitution was developed, as Bernoulli integral before Euler equation.But the related particular attentions of Vlasov equations appeared before the generalequation was developed.

Enormous attention was paid to the the Maxwell-Boltzmann distribution, whengα(E)= Aαe−βE. This distribution gives1U = eU equation for electrons, when n= 1in equation (4.1.2) and 1U = e−U for gravitation (Max Von Laue, Nobel Prize win-ner). The monoenergetic case, when gα is just a δ-function of energy, also was con-sidered as a special case. For example, Lengmuir and Boguslavskij were describing adiode in such manner; see [210, 284, 319].

4.3 Boundary-Value Problem for NonlinearElliptic Equation

Let us consider the boundary-value problem for equation (4.2.1) with function (4.2.2):{1U = ψ(U),

U|∂D = U0.(4.3.1)

It is known that the problem (4.3.1) is well posed when ψ ′(U)≥ 0; see [166] fordetails.

The uniqueness of solutions is obtained quite simply by the following reasoning: U1and U2 are two different solutions. Then1(U1−U2)= ψ(U1)−ψ(U2). Multiplyingby U1−U2 and integrating over domain D, one obtains∫

D

(U1−U2)1(U1−U2)dx=∫D

(ψ(U1)−ψ(U2))(U1−U2)dx. (4.3.2)

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Energetic Substitution 31

The right side of the equality (4.3.2) is nonnegative, since the ψ(U) function is amonotonic one. Applying the Green formula [303] to the left side, we get

∫D

g1hdx=−∫D

(∇g,∇h)dx+∫∂D

g∂h

∂ ndS.

Applying it for g= h= U1−U2, we see that taking U1−U2|∂D = 0 makes theexpression on the left nonpositive. Hence, both parts of (4.3.2) are equal to zero,∇(U1−U2)= 0 and U1−U2 = C.

Taking into account the boundary condition U1−U2 = 0, we obtain U1 ≡ U2 indomain D.

1. The boundary-value problem (4.3.1) has a unique solution for 9 ′ ≥ 0. A similar reasoningis also used for a periodical problem.

2. The periodical problem

{1U =ψ(U),

U(X+T)=U(X), T = (T1,T2,T3)

has an unique solution for 9 ′(S)≥ 0. If 9(a)= 0, for any number a ∈ R, then the solutionU(x)≡ a is known.

Exercise 4.1. Prove that, if 9(U) does not vanish in any point, then there are noperiodical solutions.

Hint: Consider the maximum or minimum point of function 9(U), depending onit’s sign.

4.4 Verifying the Condition 9 ′ ≥ 0

The condition 9 ′ ≥ 0 is a key condition, thus it has to be considered in details[284, 286]. We have:

9(S)=−4πn∑1

e2α

∫Rd

g′α

(mαv2

2+ eαS

)dv. (4.4.1)

Thus, if g′α < 0, then 9 ′ ≥ 0. Exponent type distributions, such as Maxwell-Boltzmann, for example, satisfy this condition.

Let us see what will happen when gα(E) is not monotonic. Integrating (4.4.1) inparts using spherical coordinates and denoting the area of d− 1-dimensional sphereas Bd, V = |v|, we obtain

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9 ′(S)=−4πn∑1

e2αBd

∞∫0

g′α

(mαV2

2+ eαS

)Vd−1dV =

=−4πn∑1

e2αBd

∞∫0

Vd−1

mαVdgα =

=−4πn∑1

e2α

Bd

mα

gα Vd−2∣∣∣∞V=0− (d− 2)

∞∫0

gVd−3 dV

== 4π

n∑1

e2α

Bd

mα

(d− 2)∞∫0

gαVd−3dV > 0, d > 2,

gα(eαV)≥ 0, d = 2.

All of these calculations are valid if gα(E)≥ 0 are continuously differentiable and

g′α

(mαV2

2+ eαS

)Vd−1 <

C

V1+ε

when V > R for some constants C > 0, R> 0 and ε > 0. We see that the expression9 ′(S) is nonnegative for d ≥ 2 when gα(E)≥ 0.

Assuming d = 1, we cannot use the same reasoning, because the lower limitinvolves the infinite value while integrating by parts. To be specific, the inequalitygα(E)≥ 0 is violated for some nonmonotonic functions gα(E). The most important arethe monoenergetic functions, when gα(E)= Aαδ(E−Eα). That allows the possibilityof obtaining nontrivial periodic solutions, usually called Bernshtein-Green-Kruskalwaves.

Exercise 4.2. Prove the nonmonotonicity of function 9 when gα(E)= Aαδ(E−Eα).Obtain such periodical solutions [8, 286].

Exercise 4.3. Obtain a solution of the Lengmuir diode problem.Hint: How are the electron density and catode–anode potential related? When will

the electric current flow? (The solution can be found in [210, 319]).

4.5 Conclusions

1. Taking monotonically decreasing distribution functions, one has monotonic dependence ofthe density distribution function 9(U) from the potential: 9 ′(U)≥ 0. Hence, the periodicalsolutions are absent, and any boundary-value problem has unique solution for any velocitydimension space. The main example is obtained for gα(E)= Aαe−βαE, i.e., the Boltzmann-Maxwell distribution function. In the case of gravitation, a sign of inequality is inverse andthe boundary-value problem is ill-posed and has no physical sense.

2. Since the inequality 9 ′(U)≥ 0 becomes valid for d ≥ 2, then the boundary-value problem(4.4.1) is also correct, and periodical solutions are absent.

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Energetic Substitution 33

3. For d = 1, the monotonicity of function9(U) is violated for nonmonotonic functions gα(E).In particular, for monoenergetic distributions gα(E)= Aαδ(E−Eα), there exist periodicalsolutions.

Remark 4.1. In case of energetic substitution, the trajectories of particles depend onlyfrom the initial point and thus define dynamic system. Regarding nonstationary Vlasovequations, the trajectories of particles depend from overall configuration and do notdefine a dynamic system, because a self-consistent potential U(x, t) is time dependent.

As a final note, we should say that paper [16] proved the nontrivial existence theo-rems; paper [189] investigated several quantizations of the Vlasov equation. For thefurther theory development, we recommend paper [247] in which the generalizationsof energetic substitution have been investigated for the Vlasov-Maxwell equation.

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5 Introduction to the MathematicalTheory of Kinetic Equations

5.1 Characteristics of the System

In this subsection, we briefly turn our attention to the application of classical char-acteristic theory (Bouchut [54], Hartman [129]) to transport an equation of the generalform

∂tu+ a(t,x) · ∇xu= 0, 0< t < T, x ∈ RN (5.1.1)

with a smooth coefficient

a : (0,T)×RN→ RN

and also to transport equation of divergence type

∂tf + divx[a(t,x)f ]= 0, 0< t < T, x ∈ RN . (5.1.2)

Definition 5.1. A function X(s) ∈ C1 in an interval R with values in RN , satisfyingequation

dX

ds= a(s,X(s)), (5.1.3)

is said to be characteristic for (5.1.1) or (5.1.2).

Theorem 5.1 (see [129] for details). Let a ∈ C([0,T]×RN), ∇xa ∈ C([0,T]×RN)

and

∀t ∈ [0,T] ∀x ∈ RN|a(t,x)| ≤ k(1+ |x|).

Then for all t ∈ [0,T] and x ∈ RN there exists unique characteristic on [0,T], sat-isfying X(t)= x and denoting by X(s, t,x). Here

X ∈ C1([0,T]s× [0,T]t×RNx ),

∂s∇xX, ∇x∂sX exist and coincide in C([0,T]s× [0,T]t×RNx ). If a, ∇xa ∈

Ck−1([0,T]×RN), k ≥ 1, then X ∈ Ck([0,T]s× [0,T]t×RNx ).


http://dx.doi.org/10.1016/B978-0-12-387779-6.00005-3

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Proposition 5.1. Assume conditions of Theorem 5.1 are fulfilled. Then

(i) ∀t1, t2, t3 ∈ [0,T], ∀x ∈ RN ,

X(t3, t2,X(t2, t1,x))= X(t3, t1,x). (5.1.4)

(ii) For all s, t ∈ [0,T] mapping x→ X(s, t,x) is C1 diffeomorphism in RN with inverse map-ping X(t,s, ·).

(iii) Jacobian J(s, t,x)= det(∇xX(s, t,x)) satisfies equation

∂J

∂s= (divxa)(s,X(s, t,x))J, J > 0.

Proof. A property (i) follows from the definition of mapping X and the uniquenessof the Cauchy problem; (ii) follows from (i) for t1 = t3. Denoting by I(A) conjugatecomatrix A, one obtains

∂J

∂s= tr(I(∇xX(s, t,x))∂s∇xX(s, t,x))=

= tr(I(∇xX(s, t,x))∇x[a(s,X(s, t,x))])=

= tr(I(∇xX(s, t,x))∇xa(s,X(s, t,x))∇xX(s, t,x))=

= tr(∇xa(s,X(s, t,x))∇xX(s, t,x)I(∇xX(s, t,x))=

= det∇xX(s, t,x)tr(∇xa(s,X(s, t,x)))=

= J(divxa)(s,X(s, t,x)).

Since J(t, t,x)= 1, we obtain J > 0, which proves (iii).

Remark 5.1. If divxa= 0, then J ≡ 1 and mapping X(s, t, ·) preserves the measure(Liouville theorem). In that case, equations (5.1.1) and (5.1.2) coincide, and X induces“incompressible” flow.

One of the properties of the flow X(s, t,x) states that it can described as a differen-tial equation in variable s by means of (5.1.3) written as

∂sX(s, t,x)= a(s,X(s, t,x)) (5.1.5)

and as a partial-differential equation in terms of variables (t,x).

Proposition 5.2. A flow X is described by equation

∂tX(s, t,x)+ (a(t,x) · ∇x)X(s, t,x)= 0. (5.1.6)

Proof. Differentiating (5.1.4) in t2, we obtain

∂tX(t3, t2,X(t2, t1,x))+∇xX(t3, t2,X(t2, t1,x))∂sX(t2, t1,x)= 0.

Using (5.1.5) and the property (ii) of the proposition 5.1, we obtain (5.1.6) after anevident change of variables.

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Introduction to the Mathematical Theory of Kinetic Equations 37

Theorem 5.2. (i) If u0∈ C1(RN), then there exists a unique solution u ∈ C1([0,T]×RN) of

(5.1.1) with initial value u(0,x)= u0(x) of the form

u(t,x)= u0(X(0, t,x)).

(ii) Let a,∇xa ∈ C1. If f 0∈ C1(RN), then there exists a unique solution f ∈ C1([0,T]×RN) of

(5.1.2) with initial value f (0,x)= f 0(x) of the type

f (t,x)= f 0(X(0, t,x))J(0, t,x). (5.1.7)

Proof. To prove the first part of the theorem, we multiply (5.1.6) by ∇xu0(X(0, t,x)).Therefore, if u ∈ C1 satisfies (5.1.1), then for any (t0,x0) one has

d

ds[u(s,X(s, t0,x0))]= ∂tu(s,X(s, t0,x0)) +

+∇xu(s,X(s,X(s, t0,x0)))a(s,X(s, t0,x0))= 0

and u(s,X(s, t0,x0))= u0(X(0, t0,x0)). Putting X(s, t0,x0)= x, one obtains

u(s,x)= u0(X(0, t0,X(t0,s,x)))= u0(X(0,s,x)).

Assuming f to be a solution of (5.1.2), we can prove the second statement. Namely,

d

ds[f (s,X(s, t0,x0))] = ∂tf (s, t0,x0)+∇xf (s,X(s, t0,x0))a(s,X(s, t0,x0))=

= −(divxa)(s,X(s,X(s,X(s, t0,x0))))f (s,X(s, t0,x0))=

= −f (s,X(s, t0,x0))∂sJ(s, t0,x0)

J(s, t0,x0).

Hence,

f (s,X(s, t0,x0))= f 0(X(0, t0,x0))J(0, t0,x0)

J(s, t0,x0).

Differentiating (5.1.4) in x and opening determinant, one obtains

J(t3, t2,X(t2, t1,x))J(t2, t1,x)= J(t3, t1,x).

Taking t3 = 0, t2 = s, t1 = t0, x= x0, we have

J(0, t0,x0)

J(s, t0,x0)= J(0,s,X(s, t0,x0)),

which induces (5.1.7). Or, vice versa, function f defined by (5.1.7) satisfies theequation

(∂tf + divx(a(t,x)f ))(s,X(s, t0,x0))= 0,

for all (t0,x0), which implies (5.1.2).

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Remark 5.2. Taking f 0= 1 in the preceeding equation, we have

∂tJ+ divx(a(t,x)J)= 0.

Proposition 5.3. We assume that a,∇xa ∈ C1. If f 0∈ L1(RN), then there exists a

unique solution f ∈ C([0,T],L1(RN)) of (5.1.2) with f (0,x)= f 0(x) such that f isdefined by (5.1.7).

Proof. If f 0∈ C∞0 (R

N), then it follows from (5.1.7) that f ∈ C([0,T],L1(RN)). Thenfor f 0

∈ L1(RN) there exists a sequence f 0n ∈ C∞0 such that f 0

n → f 0 in L1. Sincefn(t, ·)→ f (t, ·) is uniform in a time scale for L1, then f ∈ C([0,T],L1) and the limitin sense of distributions (weak limit) shows that f satisfies (5.1.2).

To prove this uniqueness, we assume that f ∈ C([0,T],L1(RN)) satisfies (5.1.2) interms of distributions. We define the function

g(t,x)= f (t,X(t,0,x))J(t,0,x). (5.1.8)

Then g ∈ C([0,T],L1(RN)) ( f is approximated in C([0,T],L1(R)) by a sequence ofsmooth functions fn) and, in terms of distributions, one has for g

∂tg= [∂tf (t,X(t,0,x))+∇xf (t,X(t,0,x))∂sX(t,0,x)]J(t,0,x) +

+ f ∂sJ(t,0,x)= [∂tf + divx(af )](t,X(t,X(t,0,x)))J(t,0,x)= 0.

Thus g(t,x)= f 0(x). Inverting the relation (5.1.8), we see that f is defined by formula(5.1.7).

Proposition 5.4. Let X1(s), . . . ,Xp(s) be the function satisfying the system

dXj

ds= a(s,Xj(s)),

λ1, . . . ,λp ∈ R and we define the function

f (t,x)=∑

j

λjδ(x−Xj(t)).

Then f satisfies equation in terms of distributions

∂tf + divx(af )= 0 and [0, t]×RN .

Proof. Let φ ∈ C∞0 ([0,T]×RN). Then we obtain the following chain of derivations,from which follows the result of the proposition:

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< ∂tf ,φ >=−< f ,∂tφ >=−∑

j

λj

T∫0

∂tφ(t,Xj(t))dt =

=−

∑j

λj

T∫0

[d

dtφ(t,Xj(t))−∇xφ(t,Xj(t))xj(t)

]dt =

=

∑j

λj

T∫0

∇xφ(t,Xj(t))a(t,Xj(t))dt =< af ,∇xφ >=

=−< divx(af ),φ >.

5.2 Vlasov-Maxwell and Vlasov-Poisson Systems

The Vlasov equation describes the evolution of the system of particles in the force fieldF(t,x,p), which depends on time t, position x, and momentum p. For every particlewith index j, we can write motion equations

xj = v(Pj), pj = F(t,Xj,Pj),

where Xj denotes a position j-th particle and Pj its momentum. In general, v is a func-tion of momentum; in a classical case, one obtains

v(p)=p

m. (5.2.1)

Defining the density of particles in the form f (t,x,p)dxdp in the phase space (x,p) ∈RN×RN for any fixed t as

f (t,x,p)dxdp=∑

j

δ(x−Xj(t),p−Pj(t)).

It follows from Proposition 5.4 that the function f must satisfy the Vlasov equation:

∂tf + divx(v(p)f )+ divp(F(t,x,p)f )= 0 and Rt×RNx ×RN

p . (5.2.2)

Between the infinite number of particles we look for “smooth” solutions f of (5.2.2);therefore, we may claim that function at least f ∈ L1(RN

x ×RNp ).

In this case, when the particles are subjected to collisions, we should add a nonlin-ear term into the Vlasov equation (5.2.2). Moreover, it is transformed to a nonlinearkinetic equation with collision operator Q( f ) in the right part

∂tf + divx(v(p)f )+ divp(F(t,x,p)f )= Q( f ).

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Depending on the structure of the collision operator Q( f ), this equation is said to be aBoltzmann or Fokker-Plank-Landau equation.

Let us consider a collisionless case in which the model describes the evolution ofparticles in a selfconsistent electromagnetic field. In this case, systems simulating thedescribed process are constructed in the following way: we define particle density ρand current density j by means of

ρ(t,x)= q∫

f (t,x,p)dp, j(t,x)= q∫

v(p)f (t,x,p)dp,

where q—charge of the particle. Force F is given as

F(t,x,p)= qE(t,x), E(t,x)=−∇xφ(t,x)

for a Vlasov-Poisson (VP) system (5.2.2)

−4xφ = ρ, (5.2.3)

or in the form (Lorentz force)

F(t,x,p)= q(E(t,x)+ v(p)×B(t,x))

for a Vlasov-Maxwell (VM) system (5.2.2)

∂tE− c2rotB=−j,

∂tB+ rotE = 0, (5.2.4)

divxE = ρ, divxB= 0.

In a relativistic case, the velosity v in VM system is given by the formula

v(p)=p/m√

1+ |p|2/m2c2.

In a VP system, which is formally the classical limit (c→∞) of a VM system, veloc-ity is defined by (5.2.1).

5.3 Weak Solutions of Vlasov-Poisson and Vlasov-MaxwellSystems

By a weak solution, we mean a solution in sense of distributions. In that case, thefield of force is not smooth enough to apply a classical characteristic theory describedin subsection 5.1. The proof of existence of weak solutions for VP and VM systemsis very complicated and a group of outstanding mathematicians (see Arsen’ev [17],DiPerna, P. Lions [92], and Illner, Neunzert [148]) have tried to resolve it. Here we

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present (without proof) the principle existence theorems of weak solutions of VP andVM systems obtained by Arsen’ev, DiPerna and P. Lions, and Horst. We note thatthe existence of global (week) solutions for a three-dimensional VM system has beenproven by DiPerna and P. Lions.

Proposition 5.5. For a VP system (5.2.2), (5.2.3) one has the following relation:

∂t

(∫|p|2

2mfdp+

|E|2

2

)+ divx

(∫|p|2

2mv(p)f dp+φ( j+ ∂tE)

)= 0

and conservation principle of total energy

E(t)≡∫ ∫

|p|2

2mf dxdp+

∫|E|2

2dx.

Proposition 5.6. For a VM system (5.2.2), (5.2.4) the following integral relationholds:

∂t

(∫mc2(γ (p)− 1)f dp+

|E|2+ c2|B|2

2

)+

+ divx

(∫mc2(γ (p)− 1)v(p)f dp+ c2E×B

)= 0

with

γ (p)=√

1+ |p|2/m2c2,

moreover, one has a conservation law of total energy

E(t)≡∫ ∫

mc2(γ (p)− 1)f dxdp+∫|E|2+ c2

|B|2

2dx.

Theorem 5.3. We suppose that N ≥ 3 and let f 0∈ L1⋂L∞(RN

x × ×RNp ), f 0

≥ 0,

|p|2f 0∈ L1(RN

x ×RNp ),

x

|x|N∗ f 0(x,p)dp ∈ L2(RN

x ).

Then there is a solution of the VP system (5.2.2), (5.2.3) f ∈ C([0,∞],L∞(RN

x ×RNp )−w∗) with initial value f 0, satisfying for any t ≥ 0

|| f (t, ·)||Lpxp≤ || f 0

||Lpxp, 1≤ p≤∞,

E(t)≤ E(0).

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Theorem 5.4. Let f 0∈ L1⋂L∞(RN

x ×RNp ), f 0

≥ 0, E0,B0 satisfy the agreementconditions

divB0= 0, divE0

= q∫

f 0dp,

and we assume that

|p| f 0∈ L1(RN

p ×RNp ), E0,B0

∈ L2(RNx ).

Then there is a solution of VM system (5.2.2), (5.2.4) f ∈ C([0,∞]),L∞(RNp ×RN

p )−

w∗) and E,B ∈ C([0,∞], (RNx )−w) with initial value f 0,E0,B0 and satisfying for

any t ≥ 0

|| f (t, ·)||Lpxp≤ ||f 0

||Lpxp, 1≤ p≤∞,

E≤ E(0).

5.4 Classical Solutions of VP and VM Systems

The existence of classical (smooth) solutions of the VP system (5.2.2), (5.2.3) can beestablished via two methods. The first method (see Batt [22], Horst [143], Rein [235])consists of transformation of problem and getting decay estimation of functionf (t,x,p) at |p| →∞. The second one implies direct obtaining of that decay startingfrom a priori estimations. To realize this goal, there are two main methods: e.g., theP. Lions and B. Perthame method based on dispersion estimation and the characteris-tic method developed by Pfaffelmoser [229]. For the first time, Pfaffelmoser provedglobal solvability of Cauchy problem for a three-dimensional VP system with an arbi-trary initial value. These results were simplified and improved by Horst and Schaeffer.

For a VM system, at present, the situation is the following: the first step ofreduction has been obtained by Glassey and Strauss [111], but decay estimations invelocity p remain an open problem. The existence of global solutions of VM sys-tem for dimension 2.5 (2 dimensions in x and 3 in p) was proved by Glassey andSchaeffer [115]. Existence of global classical solutions of the VM system for 3 dimen-sions still remains an open problem.

5.5 Kinetic Equations Modeling Semiconductors

A distinctive peculiarity of mathematical analysis of semiconductors is its connec-tion with hierarchy of models of transport of charged particles in various mediums.It is explained by that transport processes have a practical interest for various scalesof length and time generated by various physical effects. In particular on the verysmall scales using in microelectronic technology, quantum and/or kinetic effects areimportant.

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The Boltzmann equation for semiconductors simulates the flow of charged elec-trons in semiconductor crystals. It describes the evolution of distribution functionf = f (x,k, t) in the phase space. Here x ∈ R3—state variable, k ∈ B—wave vector,B denotes Briullen zone, connected with corresponding crystal chain. A semiclassi-cal Boltzmann equation is written in the form

∂tf + v(k) · ∇xf +q

h∇xV · ∇kf = Q(f ), x ∈ R3, k ∈ B, t > 0,

f (k,k,0)= fi(x,k), x ∈ R3, k ∈ B.

Physical constants q and h denote elementary charge and reduced Plank con-stant respectively, v(k)=(1/h)∇kε(k)—velocity of electron, ε(k)—energy and V =V(x, t)—electrical potential. Collision operator Q(f ) simulates a short (by scale oflength) interaction of electrons with crystal inclusions, phonons and electrons. Elec-trical potential V is the given function or selfconsistently closed via Poisson equation

ε4V = q(n−C(x)), (5.5.1)

where ε denotes conductivity of semiconductor and C = C(x) simulates given an ionbackground. The density of electron is defined by

n=∫B

f dk.

For modeling super-small electronic devices, we need to consider quantum effects.The quantum Boltzmann equation is a kinetic equation describing evolution the Vignerfunction w= w(x,v, t)

∂w

∂t+ v · ∇xw+

q

m∗2h[V]w= Qh(w), x,v ∈ R3, t > 0,

w(x,v,0)= w0(x,v), x,v ∈ R3.

Here Qh(w)—collision operator, V—effective potential, satisfying the Poisson equa-tion (5.5.1). Electronic density and current density are defined by formulas

n=∫R3

w(x,v, t)dv, jn =−q∫R3

vw(x,v, t)dv.

Operator 2h[V]w is pseudodifferential operator of special form.In the semiclassical limit h→ 0, a quantum Boltzmann equation is transformed to

a quantum Vlasov equation

∂w

∂t+ v · ∇xw+

q

m∗∇xV · ∇vw= 0, (5.5.2)

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which is closed by a Poisson equation (5.5.1) for potential V . The system (5.5.2),(5.5.1) is a nonlinear, nonlocal, integro-differential system. In the last 2–3 years, manyleading mathematicians both with point of view of existence problem of solutionsand from an applied point of view (semiconductor modeling) have considered thisproblem.

Transport modeling of electrons in nanostructure, for example, resonance tunneldiodes, superchains leads to quantum models: Schrodinger and Vigner equations. Gen-erally quantum effects exhibit in the local domains of devices, and in the other part(global domain) are subjected to classical equations. In this case, there arises the ques-tion: how to correctly choose the boundary conditions for quantum models and adjustthem with classical kinetic models—Vlasov or Boltzmann equations? The answer iscontained in construction of new composed classico-quantum models with transitionboundary conditions between quantum and classical domains. As an example of suchmodels, consider a one-dimensional nonstationary Vlasov-Vigner model (Ben Abdal-lah, Degond, Gamba [37]).

Let a< b—two real numbers, V(x)—smooth stationary potential on R. Then, inthe classical domain, the motion of electrons is described by a Vlasov equation

∂f

∂t+ p

∂f

∂x−

dV

dx

∂f

∂p= 0; x ∈ [a,b], t ∈ R, t ∈ R,

and, in a quantum zone, the equation of matrix density

ihρt = (Hx−Hx′)ρ, (x,x′, t) ∈ R3,

where ρ = ρ(x,x′, t)—matrix density; H =− h2

2∂2

∂x2 +V(x). Boundary interaction ofclassical domain and open quantum system is realized by means of particles incom-ing from exterior (classical domain) in the previous time interval and outcoming fromquantum domain in the next interval. We denote by 2− = ({a}×R∗+)

⋃({b}×R∗−)

the part of boundary corresponding to incoming particles and g(x,p, t)—distributionfunction on the boundary. Here electrons are quantum in [a,b] and, in [0,a]

⋃[b,L],

behave as classical particles. Let fC(x,p, t)—distribution function in the classical zoneand fQ—Vigner function in quantum zone. Then the classical distribution functionis calculated by means of solution of boundary value problem for Vlasov or Boltz-mann equation in classical zone with additional conditions of surface (boundary)“interchange” at x= a or x= b

fC(a,p, t)= Rh−pfC(a,−p, t− τ h

R(−p))+Thp fC(b,p, t− τ

ht (p)), p< 0,

fC(b,p, t)= Rh−pfC(b,−p, t− τ h

R(−p))+Thp fC(a,p, t− τ

ht (p)), p> 0.

If fC is found, then fQ is calculated as Vigner transformation for matrix of densityρ = ρ(x,x′, t).

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5.6 Open Problems for Vlasov-Poisson andVlasov-Maxwell Systems

In this section is given the list of the open problems for the VP and VM equations,kindly sent to the authors by Professor Jurgen Batt.

Problem 1. One way to construct a stationary spherically symmetric solution of the VPsystem is described in [23].

A second method is the so-called inversion method: given a function h(u,r) such that theequation

1

r2(r2u′(r))′ = h(u(r),r)

is solvable, that is, has a solution u= u(r), one defines ρ(r)= h(u(r),r) and has to solve theproblem to find a function ϕ = ϕ(E,F) with ϕ(E,F)= 0 for E < a := u(∞) such that

h(u,r)=2π

r

a∫u

2r2(E−u)∫0

ϕ(E,F)√2r2(E− u)−F

d(E,F). (5.6.1)

This equation then guarantees

ρ(|x|)= 4π∫R3

ϕ

(v2

2+ u(|x|),x2v2

− (xv)2)

dv,

v2

2+ u(|x|)=: E, x2v2

− (xv)2 =: F,

that is, the above defined ρ is the local density of the distribution function

f := ϕ(E(x,v),F(x,v)).

The substitution ξ = a− u, η = 2r2 (one-dimensional variables) gives (5.6.1) in the form

h

(a− ξ,

√η

2

)=

1

π√η

ξ∫0

η(ξ−s1)∫0

2√

2π2ϕ(a− s1,s2)√η(ξ − s1)− s2

ds2ds1

or

g(ξ,η)=1

π√η

ξ∫0

η(ξ−s1)∫0

f (s1,s2)√η(ξ − s1)− s2

ds2ds1

in R+×R+ with

h

(a− ξ,

√η

2

)=: g(ξ,η), 2

√2π2ϕ(a− s1,s2)=: f (s1,s2).

And, the problem is, when g is given, to find distribution f .

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The substitution s= s1 and t = s1+s2η

gives

g(x,y)=1

π

x∫0

t∫0

f (s,y(t− s)) ds√

x− tds dt, ξ =: x, η =: y.

Abel’s integral equation

g(x,y)=1

π

x∫0

ψ(t)√

x− tdt,

where

ψ(t) =

t∫0

f (s,y(t− s))√

x− tdt

has the solution

ψ(x)=1√π

d

dx

x∫0

g(s,y)√

x− sds.

The problem remains, given ψ , to solve (for f ) the equation

ψ(x)=

x∫0

f (s,y(x− s))ds

or

ψ(x)=∫

2(y)a

f (a)da1√

1+ y2=<f (2(y),p(x,y))

1√1+ y2

,

where < is the Radon transform in

2(y)=

(y√

1+ y2,

1√1+ y2

), p(x,y)=

xy√1+ y2

.

Notation in F. Natterer [214].Problem 2. In connection with problem 1 in [145], Hunter and Qian have extended theinversion method to cylindrically symmetric stationary solutions on a formal level. Anembedding of their arguments in a rigorous mathematical frame is needed.Problem 3. A difficult open problem is the global existence of classical solutions to theVlasov-Maxwell system for general sufficiently smooth initial data. R. Glassey, J. Schaf-fer have settled this existence problem for a 2D version of the VM system in 1998, for a2 1

2 D version in 1997 and a 1 12 D version in 1990 [114]. In the latter paper, some regularity

questions are still open. But we still have dificulties in the general existence theorem forthe classical intial value problem of the VP system (for a mathematical development of thistheorem see G. Rein [237]).

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Problem 4. To extend existence theorem for classical global solutions to the initial valueproblem of the VP system to singular initial values.

Many stationary solutions with spherical symmetry are examples of (trivially globallyexisting) singular solutions of the VP system, e.g., certain polytropes (corresponding to theM-solutions of the Emden-Fowler equation) and certain Camm-models (corresponding tothe M-solutions of the Matukuma equation). The semiexplicit spherically symmetric solu-tions constructed by R. Kurth [167] are singular nonstationary global solutions.Problem 5. To extend the above mentioned existence theorem to initial values with infinitemass.

Nonstationary solutions with infinite mass have been constructed in the paper [24]. A firsteffort to deal with initial values with infinite mass is made by S. Caprino [63]; see also [60].Problem 6. In the above-mentioned paper [24], the existence of time-periodic solutionswas proven; the semiexplicit solutions of R. Kurth [167] are also partially time-periodic. Ageneral result for time-periodic solutions, however, is still outstanding.

The question is: Which initial values lead to time-periodic solutions?It might be helpful to look into the paper by P.E. Zhidkov [320].

Problem 7. In the paper by J. Batt [22], it was proven that initial values with sphericalsymmetry and with bounded v-support lead to solutions with time-global bounded v-support.The growth of the v-support of a solution f of the VP system is estimated by the function

hη(t) := sup{|v(0, t,x,v)− v| x,v ∈ R3}.

E. Horst showed that hη(t)= O(t1+δ), δ > 0. J. Batt and G. Rein proved hη(t)= O(t2) inthe x-periodic case. For cylindrically symmetric solutions the estimate is not known. Sharpestimates found in [321] gave an example of a uniquely determined weak solution of the VPsystem whose (x,v)—support is uniformly bounded for all times.

t

y x

1

s

In general, the uniqueness of weak solutions to an initial value problem is open, for thetotal energy one only knows E(t)≤ E(0) (instead of equality). The paper by Robert R. [239]seems to be a further attempt in this direction. A.J. Majda, G. Majda and Y. Zheng [182] haveproven the nonuniqueness of weak (rather singular) solutions. The exact border betweennonuniqueness and uniqueness in the field of weak and strong solutions is not known.E. Horst and R. Hunze [141] have developed a concept to get weak solutions for the VPsystem, which has become a guideline to handle other cases (Vlasov-Poisson-Fokker-Plank,flat case of the VP system).Problem 8. In article [28], an elementary proof for the approximability of the solutions ofthe unmodified VP system by the mass-point solutions of modified N-body problems whenN→∞ and the modification parameter δ→ 0 in an appropiate way was given. However,this is not what the physicists call the “mean field limit”. For the modified VP system, themean field limit has been proven in the now classical paper of W. Braun and K. Hepp [56]“The Vlasov dynamics and its fluctuations in the 1

N limit of interating classical particles.”

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Problem 9. The same as for problem 8, but for the Vlasov-Maxwell system. E. Horst hasshown the global existence for classical solution of the modified VM system, where modifi-cation means that the current density j is replaced by the convolution j(·, t) ∗ δ, where δ is amember of a canonical sequence of δ-functions.Problem 10. The “flat” VP system is obtained, if x,v ∈ R3 are replaced by x,v ∈ R2 andin the integral form of Poisson‘s equation, i.e., in the formula for the potential u and in thedefinition of the local density ρ, the integrals

∫R2 over R3 are replaced by integrals

∫R2 over

R2 (over all x or all v, respectively).The question is: Can one obtain the solutions of the flat VP system from the solutions of

the VP system (in 3D) by a suitable limit process?The “jump relations” (see R. B. Guenther, J. W. Lee [124], Chapters 6–8) might play an

important role.S. Dietz in her dissertation [93], (unpublished) has proven the existence of classical solu-

tions locally, the existence of weak solutions globally.Problem 11. The unicity of classical solutions of the VP system (or of related systems) isnot fully understood.

Are two C1—solutions f1, f2 of the initial value problem with the same initial valuef 0∈ C1

c (R6) equal?Problem 12. Generalize the approach of [25] introduced for Emden-Fowler equation toanother equation.

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6 On the Family of the Steady-StateSolutions of Vlasov-Maxwell System

6.1 Ansatz of the Distribution Function and Reduction ofStationary Vlasov-Maxwell Equations to Elliptic System

Let us consider stationary VM system

v∂

∂rfi+

qi

mi

[E+

1

cv×B

]∂

∂vfi = 0, i= 1,2 (6.1.1)

rotE = 0 (6.1.2)

divB= 0 (6.1.3)

divE = 4π2∑

j=1

qj

∫R3

fj(v,x)dv (6.1.4)

rotB=4π

c

2∑j=1

qj

∫R3

vfj(v,x)dv (6.1.5)

∫R3

∫�1

fi(v,x)dxdv= 1. (6.1.6)

We will assume that q1 > 0 and q2 < 0, which means that f1(v,x) and f2(v,x)respectively are the ion and electron distribution functions. Taking into account con-dition (6.1.6), we also will find the stationary distribution functions of the form

fi(v,x)= fi(−αiv2+ϕi,vdi+ψi)= f (R,G), (6.1.7)

where R=−αiv2+ϕi, G= vdi+ψi and the corresponding electromagnetic fields

E(x),B(x) satisfying the system of equations (6.1.2)–(6.1.5). Here we assume

ϕi : R3→ R, ψi : R3

→ R, x ∈�1 ⊆ R3, v ∈ R3, αi ∈ R+ = [0,∞), di ∈ R3.


http://dx.doi.org/10.1016/B978-0-12-387779-6.00006-5

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Let us build the system of equations determining the functions ϕi, ψi. After substi-tution (6.1.7) into (6.1.1), we come to the relationships

E(x)=mi

2αiqi∇ϕi (6.1.8)

B(x)× di =−mic

qi∇ψi (6.1.9)

(E(x),di)= 0. (6.1.10)

Moreover, it follows from (6.1.8), (6.1.10) that

(∇ϕi,di)= 0. (6.1.11)

By contrast, from (6.1.9) we obtain

(∇ψi,di)= 0. (6.1.12)

Let us note that (B(x),di)= λi, where λi(x) is an arbitrary function at this moment.Thus, for definition of B(x) we need the joint solution of equations (6.1.9), (6.1.12)taking into account a fact that (B(x),di)= λi(x). Vector B(x) takes the form

B(x)=λi(x)

d2i

di−mic

qid2i

di×∇ψi, (6.1.13)

where d2i = d2

1i+ d22i+ d2

3i.The most outstanding fact is that fields E(x) and B(x), determined by means of

formulas (6.1.8), (6.1.9) does not depend on index i. Hence, the functions ϕi,ψi canbe searched in the form

ϕi = ϕi+ϕi(x), ψi = ψi+ψi(x), (6.1.14)

where ψi, ϕi are constants; functions ϕi(x),ψi(x) satisfy the relations ϕ2(x)= l1ϕ1(x),ψ2(x)= l2ψ1(x). Parameters l1, l2, with respect to (6.1.8), (6.1.9) are connected withthe relations

m1

α1q1= l1

m2

α2q2(6.1.15)

l2q1

m1d1 =

q2

m2d2. (6.1.16)

In this case, l1 =m1α2q2m2α1q1

< 0, mi > 0, αi > 0, q1q2 < 0. As it follows from (6.1.16),

vectors d1, d2 are linearly dependent. Since ϕi and ψi defined in (6.1.14) are constants,then ∇ϕi =∇ϕi(x), ∇ψi =∇ψi(x).

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On the Family of the Steady-State Solutions of Vlasov-Maxwell System 51

Substituting (6.1.7) and (6.1.8) into (6.1.4), we will obtain the equations

mi

2αiqi4ϕi(x)= 4π

2∑j=1

qj

∫R3

fj(v,x)dv. (6.1.17)

Since div[di×∇ψi]= 0 and taking into account (6.1.12), substituting (6.1.13) into(6.1.3) gives

(∇λi(x),di)= 0. (6.1.18)

Substituting (6.1.13) into (6.1.5), one obtains the system of equations

∇λi× di =mic

qidi4ψi(x)+

4π

cd2

i

2∑j=1

qj

∫R3

vfjdv.

The above system is solved if and only if functions ψi(x) satisfy the equations

−mic

qi4ψi(x)=

4π

c

2∑j=1

qj

∫R3

(v,di)fjdv. (6.1.19)

In this case

∇λi(x)=γi

d2i

di+4π

c

2∑j=1

qj

di×

∫R3

vfjdv

(6.1.20)

because of [di× dj]= 0, (i, j)= 1,2. Finally, combining (6.1.20), (6.1.18) and di 6= 0,we obtain

∇λi(x)=4π

c

2∑j=1

qj

di×

∫R3

vfjdv

. (6.1.21)

Relations (6.1.17), (6.1.19) are the desired system of elliptic equations related to thefunctions ϕi(x),ψi(x). Thus, the problem of finding the steady-state solutions of theVM system (6.1.1)–(6.1.5) came to a joint study of equations (6.1.17), (6.1.19) withconditions of orthogonality (6.1.11), (6.1.12) and normalization condition (6.1.6).

Systems (6.1.17), (6.1.19) will be studied assuming

fi(v,x)= e−αiv2+ϕi(x)+div+ψi(x). (6.1.22)

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In this case∫R3

fj(v,x)dv=

(π

αj

)3/2

eϕi+ψied2j

4αj (6.1.23)

∫R3

(v,di)fi(v,x)dv=(di,dj)

2αj

∫R3

fj(v,x)dv (6.1.24)

di×

∫R3

vfjdv= di×dj

2αj

∫R3

fj(v,x)dv, (6.1.25)

where i, j= 1,2. Since vectors d1, d2 are linearly dependent, then the right parts in(6.1.25) are equal to zero. Therefore, if fi(v,x) is defined by (6.1.22), then, due to(6.1.21), ∇λi(x)= 0. Hence, λi(x)= βi is constant in (6.1.13).

The right parts of (6.1.17), (6.1.19) are completely defined, due to (6.1.23), (6.1.24),and the last one takes the form

mi

2αiqi4ϕi = 4π

2∑j=1

qj

(π

αj

)3/2

eϕj+ψjed2j

4αj (6.1.26)

−mic

qi4ψi =

4π

c

2∑j=1

qj(di,dj)

2αj

(π

αj

)3/2

eϕj+ψjed2j

4αj , (6.1.27)

where i= 1,2; ϕi = ϕi+ϕi(x); ψi = ψi+ψi(x),

(d1,d2)= kd21; d2

2 = k2d21; k =

α2l2α1l1

.

Rewriting (6.1.26), (6.1.27) in the vector form

4ϕ = Af (ϕ+ψ) (6.1.28)

4ψ = Bf (ϕ+ψ), (6.1.29)

where

A=

∣∣∣∣∣∣∣∣a

x1

b

x1a

x2

b

x2

∣∣∣∣∣∣∣∣ ; B=

∣∣∣∣∣∣∣∣a

2α1cy1

bk

2α2cy1a

2α1cy2k

b

2α2cy2

∣∣∣∣∣∣∣∣ϕ =

∣∣∣∣∣ϕ1

ϕ2

∣∣∣∣∣ ; ψ =

∣∣∣∣∣ψ1

ψ2

∣∣∣∣∣f (ϕ+ψ)=

∣∣∣∣ eϕ1+ψ1

eϕ2+ψ2

∣∣∣∣

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a= 4πq1

(π

α1

)3/2

ed21

4α1 ; b= 4πq2

(π

α2

)3/2

ed2

24α2

xi =mi

2αiqi; yi =−

mic

qid2i

; i= 1,2

and introducing notations

G = A+B, 8=

∣∣∣∣∣∣∣81

82

∣∣∣∣∣∣∣ , 8i = ϕi+ψi,

after summation one obtains the equation

48= Gf (8) (6.1.30)

instead of (6.1.28), (6.1.29).For studying (6.1.26) and (6.1.27), we will use the following results.

Lemma 6.1. Let8 satisfy system of equations (6.1.30). Then system (6.1.28), (6.1.29)has a solution ϕ = ϕ0+ u1, ψ =−ϕ0+ u2, where ϕ0 is an arbitrary harmonic vector-function; u1, u2 are vector functions satisfying the linear Poisson equations

4u1 = Af (8), 4u2 = Bf (8).

Moreover, if detG 6= 0, then

u1 = AG−18, u2 = BG−18.

Starting a study of (6.1.30) let us consider two cases:

1. detG 6= 0;2. detG = 0.

Lemma 6.2. If detG 6= 0, then solution of the system (6.1.30) of the form

8=

∣∣∣∣ 81+ u82+ u

∣∣∣∣corresponds the solution of algebraic system

Gf (8)= I, I =

∣∣∣∣11∣∣∣∣ (6.1.31)(

Gf (8)=−I), (6.1.32)

where function u satisfies Liouville equation

4u= eu(4u=−eu).

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Remark 6.1. An analogous result occurs in the case when G is the nonsingular matrizof the n–th order.

Lemma 6.3. If detG = 0, then a general solution λ

∣∣∣∣ c1c2

∣∣∣∣, where λ—an arbitrary con-

stant of the uniform system Gf (8)= 0 has a correspondence to the solution family ofthe system (6.1.30) of the form

8=

∣∣∣∣ 81+ u82+ lu

∣∣∣∣ . (6.1.33)

Here u satisfies the equation of the type 4u= λc3(eu− elu), l,c1,c2,c3—constants;

8i = lnλci.

Proof. Since detG = 0, then there exists a constant l such that

G =∣∣∣∣ a11 a12la11 la12

∣∣∣∣ .Therefore, substituting (6.1.33) into (6.1.30), one obtains the equation

4u= a11e81+u+ a12e82+lu,

where∣∣∣∣∣ e81

e82

∣∣∣∣∣= λ∣∣∣∣∣∣

1

−a11

a12

∣∣∣∣∣∣if a12 6= 0. Since a11e81 + a12e82 = 0, then

a11e81+u+ a12e82+lu

= λa11[eu− elu].

6.2 Boundary Value Problem

In this section we aim to the construction of the distribution functions fi(v,x), electro-magnetic fields E(x), B(x) and setting the adecuate boundary-value problems. We willconsider the distribution functions fi(v,x) and the fields E(x),B(x) corresponding toequations (6.1.22), (6.1.8), (6.1.13), where functions ϕi,ψi satisfy (6.1.26), (6.1.27),(6.1.14)–(6.1.16) and conditions (6.1.11), (6.1.12).

Let us consider two cases:

1. l2 =m1α2q2

m2α1q1, i.e., l2 = l1;

2. l2 6=m1α2q2

m2α1q1, i.e., l2 6= l1.

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The first calls to Lemma 6.3. Actually, since

detG =−ab

2cα1l1x1y1(l2− l1)

2,

then detG < 0 for l2 6= l1 and detG = 0 for l2 = l1. Thus, in the first case detG = 0.Therefore, it is possible to use Lemma 6.3. We note, due to (6.1.15), (6.1.16), l2 =l1 = l= m1α2q2

m2α1q1< 0, α2d1 = α1d2.

In this case, matrix G can be transformed as

G =[

1

x1−

1

2α1cy1

]∣∣∣∣∣ a b

la lb

∣∣∣∣∣ .Since 8=

∣∣∣∣ 81+ u82+ lu

∣∣∣∣, then (6.1.30) degenerates into one equation

4u=

(1

x1−

1

2α1cy1

)(ae81+u

+ be82+lu), (6.2.1)

moreover 81, 82 are defined from the system

G =

∣∣∣∣∣ e81

e82

∣∣∣∣∣= 0,

due to Lemma 6.3. Since we are interested in real solutions, sign ab= sign q1q2 < 0,then 81 = lnλ, 82 = ln | − λ a

b |, where λ ∈ R+ is an arbitrary parameter.Equation (6.2.1) takes the form

4u= λa11(eu− elu), (6.2.2)

where

a11 =2πq2

1

α1m1c2e

d21

4α1 (4α21c2− d2

1)

(π

α1

)3/2

, l ∈ R−.

Let x ∈�2 ⊂ R3, 0 = ∂�. We will search for a nontrivial solution of (6.2.2) satis-fying boundary condition

u|0 = 0 (6.2.3)

and relation

(∇u,d1)= 0. (6.2.4)

With respect to (6.2.4), one needs u= u(

xd11−

yd12,

yd12−

zd13

), if d1k 6= 0, (k =

1,2,3). In addition, three-dimensional problem (6.2.2)–(6.2.4) is transformed into

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a two-dimensional one. Dirichlet problem (6.2.2)–(6.2.4) has a trivial solution u= 0.Nonlinear Dirichlet problems (6.2.2)–(6.2.4) can have a small nontrivial real solutionu→ 0 for λ→ λ0 in the neighborhood of eigenvalue λ0 of linearized problem

4u= λa11(1− l)u, x ∈�2 ⊂ R3

u|0 = 0, (∇u,d1)= 0.

Assuming without loss of generality that λ0 is smallest eigenvalue, then if it is unique,in the semineighborhood of the point λ0, there exists a small nontrivial solution u→ 0at λ→ λ0(λ→−λ0).

We note that Lyapunov-Schmidt branching equation of nonlinear problem has theform L(µξ)=∇ξU(ξ,µ), µ= λ− λ0. This equation is potential for any eigenvalueof linearized problem. Therefore, any eigenvalue λ0 will be a bifurcation point. If λ0

is odd-multiple, then real solution exists at least in semi-neighborhood of the point λ0.Detailed description of the domain �2 allows us to build the asymptotic behavior ofthe corresponding branches of nontrivial solutions u of (6.2.2).

For determining the functions ϕi, ψi, we use Lemma 6.1. Knowing vector-function8, we construct functions ϕ1+ψ1 = 81+ u, ϕ2+ψ2 = 82+ lu, substituting them inthe right part of the first equation of (6.1.26). As a result, we obtain linear equation

m1

2α1q14ϕ1(x)= λa(eu

− elu)

ϕ1(x)|0 = 0, (∇ϕ1,d1)= 0.

We will find ϕ1(x) in the form ϕ1(x)=2u, where 2—constant. Then we come to theequation

2m1

2α1q14u= λa(eu

− elu),

u|0 = 0, (∇u,d1)= 0.

Assuming 2= 2α1q1am1a11

, we obtain identity, because u is a solution of (6.2.2) under

conditions (6.2.3), (6.2.4). Hence, ϕ1(x)= 81+2u+ϕ0(x), where ϕ0(x) an arbi-trary harmonic function. Since E(x)= m1

2α1q1∇ϕ1(x), then m1

2α1q1ϕ1(x) is a potential of

required electric field. Assume that the value of this potential P0 is given on the bound-ary 0, then function ϕ0(x) is defined concretely. For its determination we will obtainlinear Dirichlet boundary value problem

4ϕ0(x)= 0,

ϕ0(x)|0 = P0−m1

2α1q18.

Since ϕ1+ψ1 = 81+ u, then, having obtained ϕ1, we construct ψ1 by formula ψ1 =

(1−2)u−ϕ0(x). Knowing 8, u, ϕ1, ψ1 and using (6.1.8), (6.1.13), (6.1.22), we find

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required E(x), B(x), fi:

fi(v,x)= e−αv2+vdi+8i(x),

E(x)=m1

2α1q1∇ (2u+ϕ0(x)) ,

B(x)=β1

d21

d1−m1c

q1d21

[d1×∇ [(1−2)u−ϕ0(x)]] ,

where 81(x)= lnλ+ u, 82(x)= ln | − λ ab | + lu;

d1 ∈ R3, d2 =α2

α1d1, αi ∈ (0,∞); l=

m1α2q2

m2α1q1< 0.

Further, we consider system of Vlasov-Maxwell equations (6.1.1)–(6.1.5) withcondition

(E,n0)|0 = e0. (6.2.5)

Finding stationary distributions in the form (6.1.7), as it is was made above, we derivedto (6.1.17), (6.1.19) and (6.1.20).

Let us assume that the folowing condition holds:

A). fi(v,x)= fi(−αiv2+ vdi+ϕi+ψi)∫

R3

fi(v,x)dv<+∞,∫R3

vfi(v,x)dv<+∞.

Systems (6.1.17), (6.1.19) possess the specific symmetry expressed by the followingproperty:

Property I. The second equations in systems (6.1.17), (6.1.19) coincide with the firstones.

For the proof of Property I it suffices to take into consideration that

ϕ2(x)= l1ϕ1(x), ψ2(x)= l2ψ1(x)

and taking into account (6.1.15), (6.1.16).

Lemma 6.4. If for some function 8(x), (α ∈ R+, d ∈ R3) the conditions∫R3

f (−αv2+ vd+8(x))dv<+∞,

∫R3

vf (−αv2+ vd+8(x))dv<+∞,

are satisfied, then∫R3

vf (−αv2+ vd+8(x))dv=

d

2α

∫R3

f (−αv2+ vd+8(x))dv. (6.2.6)

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Proof. Since under change of variables v= ξ + d2α we have identity∫

R3

vf (−αv2+ vd+8(x))dv=

∫R3

ξ f

(−αξ2

+d2

4α+8

)dξ

+d

2α

∫R3

f

(−αξ2

+d2

4α+8(x)

)dξ,

then ∫R3

vif (−αv2+ vd+8(x))dv=

=

∫R3

ξif

(−αξ2

+d2

4α+8

)dξ +

di

2α

∫R3

f

(−αξ2

+d2

4α+8(x)

)dξ,

ξ ∈ R3, (i= 1,2,3).Introducing the spherical coordinates % ≥ 0, 0≤ ϕ ≤ π , 0≤2≤ 2π , ξ1 =

% sinϕ cos2, ξ2 = %2 = % sinϕ sin2, ξ3 = % cosϕ, it is easy to see that∫R3

ξi f

(−αξ2

+d2

4α+8(x)

)dξ = 0, (i= 1,2,3).

Thus,∫R3

vif

(−αv2

+d2

4α+8(x)

)dv=

di

2α

∫R3

f dξ.

Since ξ = v− d2α , then (6.2.6) is satisfied.

Lemma 6.5. Let the distribution functions f1, f2 satisfy condition A . Then system(6.1.17), (6.1.19) and (6.1.20) is transformed to the form

p4ϕ(x)= q1A1(λ1+ϕ(x)+ψ(x))+ q2A2(λ2+ l1ϕ(x)+ l2ψ(x)), (6.2.7)

℘4ψ(x)= q1A1(λ1+ϕ(x)+ψ(x))d2

1

2α1+ q2A2×

× (λ+ l1ϕ(x)+ l2ψ(x))(d1,d2)

2α2, (6.2.8)

where

ϕ(x)= ϕ1(x), ψ(x)= ψ1(x), p=m1

8πα1q1

λ1 = c11+ c21, λ2 = c12+ c22, ℘ =−m1c2

4πq1.

Functions A1,A2 are defined in (6.2.9).

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Proof. Due to Property I, the second equations in (6.1.17), (6.1.19) can be omitted.Since we have identities∫

R3

fi(−αiv2+ vdi+ϕi+ψi)dv= Ai, (i= 1,2) (6.2.9)

from Lemma 6.4, where

A14= A1(λ1+ϕ+ψ), A2

4= A2(λ2+ l1ϕ+ l2ψ),

then the first equations in (6.1.17), (6.1.19) can be written in the form of (6.2.7),(6.2.8). Since∫

R3

vfkdv=−dk

2αk

∫R3

fkdv, di× dk = 0 (i,k = 1,2),

then Lemma 6.5 is proven.

As already mentioned, during the study of systems (6.1.17), (6.1.19) two cases weredetected.

Case 1l2 = α2q2m1/α1q1m2.

Then l2 = l1, α2d1 = α1d2. System (6.2.7), (6.2.8) takes a form

ε4ϕ(x)= q1A1(λ1+ϕ+ψ)+ q2A2(λ2+ l(ϕ+ψ)), (6.2.10)

µ4ψ(x)= q1A1(λ1+ϕ+ψ)+ q2A(λ2+ l(ϕ+ψ)) (6.2.11)

l1 = l24= l, ε =

m1

8πα1q1, µ=−

α1c2m1

2πq1d21

.

Let us consider nonlinear equation

4u= a(q1A1(λ1+ u)+ q2A2(λ2+ lu)) (6.2.12)

a=2πq1

α1c2m1(4α2

1c2− d2

1), u= ϕ(x)+ψ(x).

If u∗ satisfies (6.2.12), then (6.2.10), (6.2.11) has a solution

ϕ(x)=2u∗(x)+ϕ0(x), ψ(x)= (1−2)u∗(x)−ϕ0(x),

where 2= 4α21c2/(4α2

1c2− d2

1); ϕ0 an arbitrary harmonic function. Since solutionof (6.2.10), (6.2.11) is expressed via the solution of equation (6.2.12) and harmonic

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function, then in the Case 1 VM system is reduced to a study of “resolving”equation (6.2.12). Hence, it follows:

Theorem 6.1. Let distribution functions f1, f2 satisfy condition A . Then the corre-sponding solution of system (6.1.1)–(6.1.5) can be written in the form

f1(v,x)= f1(−αv2+ vd1+ λ1+ u∗(x)), d1 ∈ R3,

f2(v,x)= f2(−αv2+ vd2+ λ2+ lu∗(x)), d2 ∈ R3, (6.2.13)

E(x)=m1

2α1q1∇(2u∗(x)+ϕ0(x)

),

B(x)=γ1

d21

d1−m1c

q1d21

[d1×∇((1−2)u

∗(x)−ϕ0)],

where u∗(x) is defined from (6.2.12); γ1, λ1, λ2 are arbitrary constants; ϕ0(x) anarbitrary harmonic function; (∇u∗, d1)= 0, (∇ϕ0(x),d1)= 0.

Let us consider solutions of “resolving” equation (6.2.12). If arbitrary constantsλ1,λ2 are connected by means of equation

q1A1(λ1)+ q2A2(λ2)= 0, (6.2.14)

then (6.2.12) has a trivial solution u∗(x)= 0. In this case, construction of nontrivialsolutions of (6.2.12) is a well-known problem about bifurcation point for nonlinearequations. For the solution of this problem we need use boundary condition (6.2.5).Due to (6.2.5), one has

∂

∂n

(2u∗(x)+ϕ0(x)

)|0 =

2α1q1

m1e0(x).

Assuming

∂

∂nu∗(x)|0 = 0,

∂

∂nϕ0(x)|0 =

2α1q1

m1e0(x),

we obtain a linear boundary value problem

4ϕ0(x)= 0,

∂

∂nϕ0(x)|0 =

2α1q1

m1e0(x), (∇ϕ0(x),d1)= 0 (6.2.15)

and a nonlinear boundary value problem

4u(x)= a(q1A1(λ1+ u)+ q2A2(λ2+ lu)), (6.2.16)

∂

∂nu(x)|0 = 0, (∇u(x),d1)= 0,

a= 2πq1(4α21c2− d2

1)/α21c2m1,

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where arbitrary constants λ1,λ2 satisfy (6.2.14). So (6.2.16) would have the nontrivialsolution u→ 0, λ→ λ∗, where λ= (λ1,λ2) satisfies (6.2.14), it is necessarily thatlinearized problem

4ϕ(x)= a[q1A′1(λ

∗

1)+ q2A′2(λ∗

2)]ϕ(x)

∂

∂nϕ(x)|0 = 0, (∇ϕ(x),d1)= 0

should have the nontrivial solution u→ 0, λ→ λ∗.

Case 2l2 6= α2q2m1/α1q1m2.

In this case, we restrict to the construction of solutions ϕ(x),ψ(x) of (6.2.7), (6.2.8)satisfying the condition

ϕ(x)+ψ(x)= l1ϕ(x)+ l2ψ(x). (6.2.17)

Let us assume that the condition is satisfied.

B). There are constants λ1,λ2, which satisfy the identity

q1A1(λ1+ u)

[4α1(1− l1)−

d21

α1c2(1− l2)

]=

= q2A2(λ2+ u)

[4α1(l1− 1)−

(d1,d2)

α2c2(l2− 1)

]for functions f1, f2 together with condition A and some u ∈ R1.

Lemma 6.6. Let condition B be satisfied. Then solution of (6.2.7), (6.2.8) with(6.2.17) has the form

ϕ(x)=l2− 1

l2− l1u∗(x), ψ(x)=

1− l1l2− l1

u∗(x),

where u∗(x) is a solution of equation

4u= bA1(λ1+ u), (6.2.18)

b=8πq2

1(l2− l1)(d21α2− (d1,d2)α1)

m1(4α1α2c2(l2− 1)− (d1,d2)(l2− 1)).

Proof. After nondegenerate variable change u1 = ϕ+ψ,u2 = l1ϕ+ l2ψ , system(6.2.7), (6.2.8) becomes

4u1 = (a1− a2)q1A1+ (a1− a3)q2A2,

4u2 = (l1a1− l2a2)q1A1+ (l1a1− l2a3)q2A2,

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where

a1 =8πα1q1

m1, a2 =

2πq1d21

α1c2m1, a3 =

2πq1(d1,d2)

α2c2m1.

The right sides of the last system are equal to each other in view of condition B. Dueto the same condition, equations over 4u1,4u2 are reduced to (6.2.18).

Theorem 6.2. Let conditions A,B, l2 6=α2q2m1

α1q1m2hold. Then system (6.1.1)–(6.1.5),

(6.2.5) has a solution

fi(v,x)= fi(−αiv2+ vdi+ λ1+ u∗(x)) (i= 1,2),

E(x)=m1(l2− 1)

2α1q1(l2− l1)∇u∗(x),

B(x)=γ1

d21

d1−m1c(1− l1)

q1d21(l2− l1)

d1×∇u∗(x),

where γ1 is an arbitrary constant. Function u∗(x) satisfies (6.2.18) with the conditions

(∇u∗,d1)= 0,∂

∂nu∗|0 =

2α1q1(l2− l1)

m1(l2− 1)e0(x).

Let us show the application of general Theorem 6.1 and Theorem 6.2.Assume, it is necessary to determine distributions

f1(v,x) = e−α1v2+vd1+81(x), d1 ∈ R3,

f2(v,x) = e−α2v2+vd2+82(x), d2 ∈ R3,

(6.2.19)

v ∈ R3, x ∈�2 ∈ R3 and the corresponding fields E, B satisfying VM system(6.1.1)–(6.1.5) with boundary condition (6.2.5). Due to Lemma 1.5, unknown func-tions 81(x), 82(x) can be occured as

81(x)= λ1+ϕ(x)+ψ(x), 82(x)= λ2+ l1ϕ(x)+ l2ψ(x).

We suppose that parameters α1,α2,d1,d2, l1, l2 satisfy conditions (6.1.15), (6.1.16).Consider two cases.

Case 1l2 = α2q2m1/α1q1m2.

Then for finding of parameters λ1,λ2 and functions ϕ(x),ψ(x), the equa-tion (6.2.12) should be resolved, and, therefore, boundary value problems (6.2.15),(6.2.16) are solved regarding Theorem 6.1. Since

A1(λ1+ u)=

(π

α1

)3/2

eλ1+

d21

4α1 eu,

A2(λ2+ lu)=

(π

α2

)3/2

eλ2+

d22

4α2 elu,

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then Condition A is satisfied. Equation (6.2.12) will have the nontrivial solution forsome λ1, if

λ2 = ln

−[α2

α2

]3/2(q1

q2

)eλ1+

14

(d21α1−

d22α2

).In this case, boundary value problem (6.2.16) is Neumann problem for equation of thetype

4u= σg(α1,d1)(eu− elu), (6.2.20)

∂

∂nu|0 = 0, (∇u,d1)= 0,

where

σ = eλ1 , g(α1,d1)= ag1

(π

α1

)3/2

ed124α1 .

Problem (6.2.20) can have nontrivial solutions u(σ )→ 0 for σ → σ ∗, only in neigh-borhood of eigenvalues σ ∗ of the problem

4u= σ ∗g(α1,d1)(1− l2)u,

∂

∂nu|0 = 0, (∇u,d1)= 0.

After obtaining eigenvalues σ ∗ (bifurcation points of (6.2.20)) and having built solu-tions u(σ )→ 0 for σ → σ ∗, we find solutions of VM system (6.1.1)–(6.1.5), (6.2.5)by formulas (6.2.13).

Case 2l2 6= α2q2m1/α1q1m2.

Then l2 6= l1 and to find λ1,λ2,ϕ,ψ , we need use Theorem 6.2. Moreover,

Ai(λi+ u)=

(π

αi

)3/2

eλi+

d2i

4αi eu.

Therefore, condition A is satisfied, moreover, condition B is reduced to B′:

q1

(π

α1

)3/2

eλ1+

d21

4α1

(4α1(1− l1)−

d21(1− l2)

α1c2

)=

= q2

(π

α2

)3/2

eλ2+

d22

4α2

(4α2(l1− 1)−

(d1,d2)(l2− 1)

α2c2

).

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Since q1q2 < 0, then from B′, one can define eλ2 , if parameters α1,α2,d1, d2, l1, l2satisfy inequility(

τ − d21ω

α1

)(τ − (d1,d2)

ω

α2

)> 0,

where τ = 4α1(1− l1), ω =1−l2

c2 . The solution of equation (6.2.18) is well-knownLiouville equation:

4u= σh(α1,α2,d1,d2)eu, (6.2.21)

∂

∂n|0 =

2α1q1(l2− l1)

m1(l2− 1)e0, (∇u,d)= 0,

σ = eλ, h(α1,α2,d1,d2)= b

(π

α1

)3/2

ed2

14α1 .

Thus, in the case of distributions of exponential form (6.2.19), the solution of(6.1.1)–(6.1.5), (6.2.5) is reduced to linear Neumann problem (6.2.15) for l2 = l1 andto the problem of bifurcation point for equation of sh-Gordon type or to Neumannproblem for Liouville equation (6.2.21), where σ ∈ R+ an arbitrary constant for l2 6= l1and to additional restriction on parameters α1,α2,d1,d2, l1, l2 included in (6.2.19).

6.3 Solutions with Norm

Let us examine constructed solutions with respect to the norm definition. With respectto normalization condition (6.1.6) and taking into account relation (6.1.23) free param-eters α1, α2, d1, λ, l must satisfy the conditions

a

4πq1e81

∫�2

eu(x)dx=1,

b

4πq2e82

∫�2

elu(x)dx=1,(6.3.1)

where 81 = lnλ, 82 = ln | − λ ab |. Therefore,

aλ

4πq1

∫�2

eu(x)dx= 1, −aλ

4πq2

∫�2

elu(x)dx= 1,

moreover, function u(x) satisfies equation (6.2.1), which takes the form

4u= λa

(1

x1−

1

2α1cy1

)(eu− elu

). (6.3.2)

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Hence, one needs to solve (6.3.2) with conditions (6.3.1). Thus, we obtain solutionswith norm. Excluding aλ from (6.3.1), we obtain equation

4u= 4π

[1

x1−

1

2α1c1y1

][q1eu(x)∫�2

eu(x)dx+

q2elu(x)∫�2

elu(x)dx

](6.3.3)

with condition∫�2

(q1elu(x)

+ q2eu(x))

dx= 0. (6.3.4)

Thus, normed solution u(x) leads to the problem (6.3.1), (6.3.2), or to the problem(6.3.3), (6.3.4). If one finds solution u∗(x) satisfying (6.3.3) and equation

4u= 4π

(1

x1−

1

2α1cy1

)(eu− elu

) q1∫�1

eu(x)dx,

then condition (6.3.4) also will be satisfied for this solution, giving us a normalizedsolution.

Let l2 6=α2q2m1

α1q1m2. Then l2 6= l1 and detG 6= 0.

In this case we study (6.1.26), (6.1.27) using Lemma 6.2 and searching for real val-ued solutions 8 of (6.1.31), (6.1.32). So, we have to define conditions when systems

G =∣∣∣∣ x1x2

∣∣∣∣= ∣∣∣∣ 11∣∣∣∣ (6.3.5)

G =∣∣∣∣ x1x2

∣∣∣∣=− ∣∣∣∣ 11∣∣∣∣ (6.3.6)

have the positive solutions x1 > 0, x2 > 0. If aij elements of matrix G, then

G−1=

1

detG

∣∣∣∣ a22 −a12−a21 a11

∣∣∣∣ ;moreover, detG < 0. Therefore, one of the systems has the positive solution, if D=(a22− a12)(a11− a21) > 0. Since

a22− a12 = b

[l1− 1

x1+

k(1− l2)

2α2cy1

],

a11− a21 = a

[1− l1

x1+

l2− 1

2α1cy1

],

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where ab< 0, l1 < 0, then

D= |ab|

[1+ |l1|

x1+

k2(l2− 1)

2α1cy1

][1+ |l1|

x1+

l2− 1

2α1cy1

].

Moreover, D> 0, if

1+ |l1|

x1>

1− l22cy1

max

{1

α1,

k

α2

},

or

1+ |l1|

x1<

1− l22cy1

min

{1

α1,

k

α2

}.

Substituting values l1,x1,y1 into the inequilities given above, we obtain

4α1c2

q1d21m2

(α1q1m2+α2|q2|m1) >max

{1− l2, (1− l2)

l2l1

}, (6.3.7)

4α1c2

q1d21m2

(α1q1m2+α2|q2|m1) <min

{1− l2, (1− l2)

l2l1

}.

Introducing notation

L(αi,qi,mi,d1)=4α1c2

q1d21m2

(α1q1m2+α2|q2|m1),

we rewrite (6.3.7) in the form

L(αi,qimi,d1) >max

{1− l2, (1− l2)

l2l1

}, (6.3.8)

L(αi,qi,mi,d1) <min

{1− l2, (1− l2)

l2l1

}. (6.3.9)

We construct straight line 1− l2 on plane yOl2 and parabola y= (1− l2)l2l1

, where l1is a given constant (see figure).

y

−2 −1−1

0

α

3

2

1

0

1 2

l2γβ

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If we fix αi, qi, mi, d1, then l1 and value L(αi,qimi,d1) in (6.3.8), (6.3.9) willbecome the specific constants. Let us draw straight line y= 1 and let us denote as α,β, γ abscissas of intersection points with parabola. Then for l2 ∈ (−∞,α), inequal-ity (6.3.9) will be satisfied, a22− a12 < 0, a11− a21 < 0, and for l2(β,γ ), inequality(6.3.8) with a22− a12 > 0, a11− a21 > 0. Since detG < 0, then system (6.3.5) has thepositive solution for l2 ∈ (−∞,α), and for l2 ∈ (β,γ ), sistema (6.3.6) has the positivesolution.

Having defined 81, 82, we build vector

8=

∣∣∣∣∣ 81+ u

82+ u

∣∣∣∣∣ ,where u satisfies a Liouville equation 4u= eu, if 81, 82 correspond to the solutionof system (6.3.5) or of equation 4u=−eu, if 81, 82 correspond to the solution of(6.3.6). For determining the functions ϕ1, ψ1, we will use Lemma 6.1. Knowing 8,we substitute it in the right side of the first equation of system (6.1.26). One obtainslinear equation

m1

2α1q14ϕ1(x)= (ae81 + be82)eu. (6.3.10)

Let, for the definition, function u(x) satisfy equation 4u(x)= eu(x), then for

2=2(ae81 + be82)α1q1

m1,

function ϕ1(x)=2u(x) satisfies (6.3.10). Thus, ϕ1(x)= 81+2u(x). Since 81 =

ϕ1+ψ1, then ψ1(x)= (1−2)u(x).We demonstrate that, here, functions fi(v,x) do not satisfy normed condition (6.1.6),

not with what values of the free parameters, if q2 =−q1. Indeed it is necessary fornorming that

a

q1e81 =

b

q2e82 =

1∫�2

eu(x)dx,

where

G

∣∣∣∣∣ e81

e82

∣∣∣∣∣=∣∣∣∣ 11∣∣∣∣ .

Therefore,

a

(1

x1−

1

2α1cy1

)e81 + b

(1

x1−

k

2α2cy1

)e82 = 1,

a

(l1x1−

l22α1cy1

)e81 + b

(l1x1−

kl22α2cy1

)e82 = 1,

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or the same(q1

x1−

q1

2α1cy1

)a

q1e81 +

(q2

x1−

kq2

2α2cy1

)b

q2e82 = 1,(

q1l1x1−

q1l22α1cy1

)a

q1e81 +

(q2l1x1−

q2kl22α2cy1

)b

q2e82 = 1.

Since aq1

e81 =bq2

e82 , then it is necessary

q1

(1− l1

x1−

1− l22α2cy1

)+ q2

(1− l1

x1− k

1− l22α2cy1

)= 0.

If q2 =−q1, then it is necessary that 1α1=

kα2

. Since k = α2l2α1l1

, then 1α1=

l2α1l1

, i.e., l1 =l2. However, in our case, l1 6= l2, providing us with contradiction. So, while l1 6= l2,the normalyzing condition (6.1.6) is fulfilled for q2 6= −q1.

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7 Boundary Value Problems for theVlasov-Maxwell System

7.1 Introduction

At present the investigation of the Vlasov equation goes in two different directions.The first direction is related to the existence theorems for Cauchy problem and usesan a priori estimation technique as the basis for research. The second one implementsthe reduction of the initial problem to a simplified one introducing a set of distributionfunction (ansatz), followed by reconstruction of the characteristics for electromagneticfields in an evident form.

This is a rather restrictive approach, since distribution function has a special form.By contrast, it allows us to solve a problem in an explicit form, which is important forapplications.

The statement and investigation of the boundary value problem for the Vlasovequation are very difficult and have only been considered in simplified cases (seeAbdallach [30], Guo [125], Degond [89]). Reducing it to the boundary value problemfor a system of nonlinear elliptic equations allows us to show a solvability in somecases. Doing the same for the initial statement of problem is not that simple.

Nevertheless, both directions are related in terms of special structures used forstudying kinetic equations. For example:

l Energy integral is applied in both cases for obtaining energy estimations in existence theo-rems and for construction of Lyapunov functionals;

l Virtual identities in stability and instability analysis in special classes of solutions of Vlasovequation.

It is known that the solution of Vlasov equation (see Vlasov [305, 306]) are arbi-trary functions of first integrals of the characteristic system (till now smoothness ofthe solutions remains a complicated unsolved problem), defining the trajectory of aparticle motion in electromagnetic field

r = V, V =qi

mi

(E(r, t)+

1

cV ×B(r, t)

), (7.1.1)

where r4= (x,y,z) ∈�2 ⊆ R3, V

4= (Vx,Vy,Vz) ∈�1 ⊂ R3—position and velocity of

a particle, E4= (Ex,Ey,Ez)—a tension of electrical field, B

4= (Bx,By,Bz)—magnetic


http://dx.doi.org/10.1016/B978-0-12-387779-6.00007-7

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induction and mi,qi—mass and charge of a particle of i-th kind. For N-componentdistribution function, the classical Vlasov-Maxwell system has the form

∂t fi+V · ∇rfi+qi

mi

(E+

1

cV ×B

)·∇V fi = 0, i= 1, . . . ,N, (7.1.2)

∂tE = c curlB− j, (7.1.3)

divE = ρ, (7.1.4)

∂tB=−c curlE, (7.1.5)

divB= 0. (7.1.6)

The charge and current densities are defined by formulas

ρ(r, t)= 4πn∑

i=1

qi

∫�1

fidV,

j(r, t)= 4πn∑

i=1

qi

∫�1

fiVdV.

(7.1.7)

We impose the specular reflection condition on the boundary for distribution function

fi(t,r,v)= fi(t,r,v− 2(vN�(r))N�(r)), t ≥ 0,r ∈ ∂�, v ∈�,

where N�(r) is a normal vector to the boundary surface.In applied problems, the impact of magnetic field is often neglected. This limit sys-

tem is known as the Vlasov-Poisson (VP) one, where the Maxwell equations degener-ates to the Poisson equation

4ϕ = 4πn∑

i=1

qi

∫�1

fidV, (7.1.8)

where ϕ(r, t)—a scalar potential of the electrical field.In a general case, the distribution function may be represented in the form

fi = fi(Hi1,Hi2, . . . ,Hil), i= 1, . . . ,N,

where Hil is the first integral (is constant along the characteristics of the equation)for (7.1.1).

In reality, it is not easy to select a structure of the distribution function which isconnected with electromagnetic potentials aiming to transform the initial system intosimplified form. Hence, in practice, one is usually restricted to the integrals of energyHi =−ci|V|2+ϕ(r, t) or H0

i =−di|V|2+ϕ(r) as in the stationary problem case (seeVlasov [305, 306]). At the same time, an introduction of the following ansatz

Hil = ϕil+ (V,dil)+ (AilV,V)+∑

m+k+j=3

ailmkjV

m1 Vk

2V j3

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Boundary Value Problems for the Vlasov-Maxwell System 71

generalizes the form of the distribution function. Here V4= (V1,V2,V3) and (AilV,V)

is a quadratic form; the following are the forms of higher degrees. In this case, matricesAil and coefficients ail

mkj should be connected with the system (7.1.2)–(7.1.6) convert-ing the first integrals for the characteristic system (7.1.1) into Hil.

The first statement of existence problem of classical solutions for the one-dimensional Vlasov equation has been given by Iordanskii [149], and the existenceof generalized (weak) solutions for the two-dimensional problem has been proven byArsen’ev [17].

The results of Neunzert [216], Horst [140], Batt [22], Illner, Neunzert [148],Ukai, Okabe [278], DiPerna, Lions P. [92], Wollman [310, 311], Batt, Rein [26],and Pfaffelmoser [229] are devoted to existence of solutions for (7.1.2) and (7.1.8).Degond [86], Glassey, Strauss [111], Glassey, Schaeffer [112]–[115], Horst [142, 143],Cooper, Klimas [76], Schaeffer [253], Guo [125], and Rein [235] concern its general-ization to the VM system (7.1.2)–(7.1.6).

Some rigorous results obtained recently (see Guo [125], Abdallach [30],Degond [89], Abdallach, Degond, Mehats [32], Vedenyapin [284], [286], Batt,Fabian [27], Braasch [55], Guo, Ragazzo [126], Dolbeault [96], Poupaud [232], Caf-farelli, Dolbeault, Markowich, Schmeiser [59], Ambroso [4]) relate to analysis of(7.1.2)–(7.1.6), (7.1.2)–(7.1.8) in the bounded domains with boundary conditions.

We have to mention that techniques used to prove the existence of solutions ofCauchy problem for the VM and VP systems for (x ∈ R3,v ∈ R3) have limited appli-cability in bounded domains. Hence, a necessity to study VM and VP systems withboundary conditions is valid. That is why, before presenting our own results, wehave to outline some already published results on VM and VP systems in boundeddomains.

7.2 Existence and Properties of the Solutions of theVlasov-Maxwell and Vlasov-Poisson Systemsin the Bounded Domains

In the case of spherical symmetry rather complete results were obtained by Batt,Faltenbacher, and Horst [23]. In the next paper by Batt, Berestycki, Degond,Perthame [24], a family of “local isotropic” solutions of nonstationary problem ofthe VP system for distribution function

f (t,r,V)=8

(W(t,r)+

(U−Ar)2

2

), U(t,r)=W(t,r)+

(Ar)2

2,

t ∈ R, r ∈ D⊂ R3, v ∈ R3, 8 : R→ [0,∞), W : R3→ R,

was constructed. Here U—potential and A—antisymmetric 3× 3- matrix. Under thisassumption, the VP system is reduced to the Dirichlet boundary value problem for the

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nonlinear elliptic equation

4W + 2|w| = 4π∫R3

8

(W +

1

2|v|2

)dv, w= (w1,w2,w3) ∈ R.

The existence of the solution for the named problem is proved using the lower—uppersolution method.

The stationary solutions of n-component VP system for distribution functiondepending on integral of energy fi(E) were studied by Vedenyapin [284], [286]. Heproved the existence of solution for Dirichlet problem

−4u(r)= ψ(u), u(r)|∂D = u0(r),

ψ(u)= 4πn∑

k=1

qk

∫R3

gk

(1

2mk|v|

2+ qku

)dv, (7.2.1)

where an arbitrary function ψ satisfies the condition (i), dduψ(u)≥ 0. Here u(r)—

scalar potential, gk(·)—nonnegative continuously differentiable functions, D⊂ R3—domain with a smooth boundary enough, u0(r)—potential given on the boundary. Ifr ∈ D⊂ Rp, v ∈ Rp, then the boundary value problem (7.2.1) has a unique solution forarbitrary nonnegative functions gk (Vedenyapin [284]).

Rein [236] proved the existence of solution of (7.2.1) by variational method undercondition (i).

In their paper [27], Batt and Fabian studied a transformation of the stationary VPsystem into (7.2.1) in general case, considering distribution functions depending onenergy fi(E) and on the sum of energy and momentum fi(E+P). Using a lower—upper solution method (Pao [224]), they proved the existence of the solutions (7.2.1)under condition (i). Therefore, the condition (i) became a primary condition to provethe existence theorems for the problem (7.2.1). The general weak global solution ofthe VP system has been presented by Weckler in [309].

Dolbeault [96] proved the existence and uniqueness of Maxwellian solutions

f (t,x,v)=1

(2πT)N/2ρ(x)e

−|v|2

2T , (x,v) ∈�×RN

using variational methods.A new direction in the study of the VP system is connected with a free bound-

ary problem for semiconductor modeling. Caffarelli, Dolbeault, Markowich, andSchmaiser [59] considered a semilinear elliptic integro-differential equation with Nue-mann boundary condition

ε4φ = q(n− p−C) – �,

∂φ

∂ν= 0 – ∂�,

(7.2.2)

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where local densities of electrons n(x) and holes p(x) in insulated semiconductor aregiven by Boltzmann—Maxwellian statistics

n(x)=N exp(qφ(x)/(kT))∫�

exp(qφ/(kT))dx, p(x)=

Pexp(−qφ(x)/(kT))∫�

exp(−qφ/(kT))dx.

C(x)—is given background, x ∈�. �⊂ Rd is a bounded domain. Using a variationalproblem statement they proved the existence and uniqueness of the solutions andshowed that the limit potential is a solution of the free boundary problem.

Concerning a study of nonlocal problem (7.2.2), we recommend the paper byMaslov [192].

7.3 Existence and Properties of Solutions of theVM System in the Bounded Domains

Changing velocity v by its relativistic analogue v= v/√

1+ |v|2 we have to faceanother complicated problem, since the classical VM system is not invariant in thesense of Galilei and Lorentz.

Adding boundary conditions

E(t,x)×N�(x)= 0, B(t,x)N�(x)= 0, t ≥ 0, x ∈ ∂�

to the system (7.1.2)–(7.1.7) we obtain a different problem statement. Here N�—vector of the unit normal to ∂� and reflection condition

fk(t,x,v)= fk(t,x, v(x,v)), t ≥ 0, x ∈ ∂�, v ∈ R3, (7.3.1)

where v : R3→ R3—bijective mapping for x ∈ ∂�. One of the most known reflection

mechanisms is a specular reflection condition of the form

v(x,v)= v− 2(vN�(x))N�(x), x ∈ ∂�, v ∈ R3

or reflection condition

v(x,v)=−v, x ∈ ∂�, v ∈ R3.

At the present, only a few number of papers study the VM system in boundeddomains. For the first time, the boundary value problem for the single dimensionalVM system has been considered by Cooper, Klimas [76].

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In the paper [248] Rudykh, Sidorov, and Sinitsyn constructed stationary classicalsolutions ( f1, . . . , fn,E,B) for the VM system of the special form (Rudykh—Sidorov—Sinitsyn ansatz)

fk(x,v)= ψk(−αkv2+µ1kU1(x),vd+µ2kU2(x)),

E(x)=1

α1q1∇U1(x),

B(x)=−1

q1d 2(d×∇U2(x)).

Here functions ψk : R2→ [0,∞) and parameters d ∈ R3

\{0},αk > 0, µik 6= 0 (k ∈{1, . . . ,n}, i ∈ {1,2})—are given; functions U1,U2 have to be defined. This approach(RSS ansatz) is closely connected with the paper of Degond [86].

Batt and Fabian [27] applied RSS ansatz technique for the VM system with dis-tribution functions ψ(E), ψ(E,F), ψ(E,F,P), where functions E(x,v), F(x,v) andP(x,v)—are the first integrals of Vlasov equation (7.1.2). Braasch in his own the-sis [55] extended RSS results to the relativistic VM system.

7.4 Collisionless Kinetic Models (Classical and RelativisticVlasov-Maxwell Systems)

In this area, existence theorems (and global stability) of renormalized solutionsin the bounded domains (when trace is defined in the boundary) were proven byMischler [205, 206]. Abdallah and Dolbeault [36] also developed the entropic meth-ods in bounded domains for qualitative study of behavior of global solutions of theVP system. Regularity theorems of weak solutions on the basis of scalar conservationlaws and averaging lemmas were proved by Jabin and Perthame [151]. Jabin [150]also obtained the local existence theorems of weak solutions of the VP system in thebounded domains. For modeling of ionic beams, Ambroso, Fleury, Lucquin-Desreux,and Raviart [5] proposed some new kinetic models with a source. Existence theoremsof global solutions of the Vlasov-Einstein system in the case of hyperbolic symmetrywere proved by Andreasson, Rein, and Rendall [6].

7.4.1 Quantum Models: Vigner-Poisson and Schrodinger-PoissonSystems

In their paper, Abdallah, Degond, and Markowich [33] considered the Child-Langmuirregime for stationary Schrodinger equation. The Authors developed a semiclassi-cal analysis for quantum kinetic equations with passage in limit h→ 0 to classicalVlasov equation with special boundary “transition” conditions from quantum zone toclassical. New results were obtained for Boltzmann-Poisson, Euler-Poisson, Vigner-Poisson-Fokker-Plank systems (like existence and uniqueness of the solutions, hydro-dynamic limits, solutions with a minimum energy and dispersion properties).

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7.4.2 Mixed Quantum–Classical Kinetic Systems

Abdallah [37] considered the Vlasov-Schrodinger (VS) and Boltzmann-Schrodingersystems for one-dimensional stationary case. Nonstationary problems for VS sys-tem with boundary “transition” conditions from classical zone (Vlasov equation) toquantum (Schrodinger equation) are studied in the paper by Abdallah, Degond, andGamba [37].

We study the special classes of stationary and nonstationary solutions of VM sys-tems. Being constructed, such solutions lead us to the systems of nonlocal semilin-ear elliptic equations with boundary conditions. Applying the lower-upper solutionmethod, the existence theorems for solutions of the semilinear nonlocal elliptic bound-ary value problem under corresponding restrictions upon a distribution function areobtained. It was shown that, under certain conditions upon electromagnetic field, theboundary conditions and specular reflection condition for distribution function aresatisfied.

7.5 Stationary Solutions of Vlasov-Maxwell System

In this section we consider the system

V ·∂

∂rfi(r,V)+

qi

mi

(E+

1

cV ×B

)·∂

∂Vfi(r,V)= 0, (7.5.1)

rotE = 0, (7.5.2)

divB= 0, (7.5.3)

divE = 4πn∑

k=1

qk

∫�1

fk(r,V)dV, (7.5.4)

rotB=4π

c

n∑k=1

qk

∫�1

V fk(r,V)dV. (7.5.5)

Here fi(r,V)—distribution function of the particles of i-th kind; r4= (x,y,z) ∈

∈�2, V4= (Vx,Vy,Vz) ∈�1 ⊂ R3—coordinate and velocity of particle respectively;

E,B—electric field strength and magnetic induction; mi,qi—mass and charge of par-ticle of i-th kind.

We shall seek stationary distributions of the form

fi(r,V)= fi(−αi|V|2+ϕi,V · di+ψi)

4= fi(R,G) (7.5.6)

and corresponding selfconsistent electromagnetic fields E and B satisfying (7.5.2)–(7.5.5). We assume that

i) fi(R,G)—fixed differentiable functions of own arguments; αi ∈ R+, di ∈ R3 are freeparameters, |di| 6= 0; ϕi = c1i+ liϕ, ψi = c2i+ kiψ , where c1i,c2i—constant; for all ϕi,ψi

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the integrals∫R3

fidv,∫R3

V fidv

are converged. Unknown functions ϕi(r), ψi(r) have to be defined in such manner that sys-tem (7.5.1)–(7.5.5) will satisfy the relation (E(r),di)= 0, i= 1, . . . ,N. The last condition isnecessary for solvability of (7.5.1) in a class (7.5.6) for ∂ fi/∂R|v=0 6= 0.

7.5.1 Reduction of the Problem (7.5.1)–(7.5.5) to the System ofNonlinear Elliptic Equations

We construct the system of equations to define the set of functions ϕi, ψi. Substituting(7.5.6) into (7.5.1) and equating to zero the coefficients at ∂ fi/∂R and ∂ fi/∂G, weobtain

E(r)=mi

2αiqi∇ϕi, (7.5.7)

B(r)× di =−mic

qi∇ψi, (7.5.8)

(E(r),di)= 0.

Here ϕi, ψi—arbitrary functions satisfying conditions

(∇ϕi,di)= 0, i= 1, . . . ,N, (7.5.9)

(∇ψi,di)= 0. (7.5.10)

Vector B is

B(r)=λi(r)

d2i

di− [di×∇ψi]mic

qid2i

, (7.5.11)

where λi(r)= (B,di)—function has to be defined. Having defined ϕi,ψi such that sys-tem (7.5.2)–(7.5.5) to be satisfied, one can find unknown functions fi,E,B by formulas(7.5.6), (7.5.7), (7.5.11).

Unknown vectors ∇ϕi, ∇ψi are linear dependent by virtue of (7.5.7), (7.5.8). Thenwe shall seek ϕi, ψi in the form

ϕi = c1i+ liϕ, ψi = c2i+ kiψ, (7.5.12)

where c1i,c2i—constants. Because of (7.5.7), (7.5.8) parameters li,ki are connectedby the following relations

li =m1

α1q1

αiqi

mi, i= 1, . . . ,N, (7.5.13)

kiq1

m1d1 =

qi

midi. (7.5.14)

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From (7.5.4) with (7.5.7), one obtains the system

4ϕi =8παiqi

mi

n∑k=1

qk

∫�1

fk(r,V)dV.

Since div[di×∇ψi]= 0, then substituting (7.5.11) into (7.5.3), one has

(∇λi(r),di)= 0. (7.5.15)

Taking into account (7.5.11), from (7.5.5) we obtain the system of linear algebraicequations for ∇λi

∇λi× di =mic

qidi4ψi+

4π

cd2

i

n∑k=1

qk

∫�1

V fkdV. (7.5.16)

To solve (7.5.16), it is necessary and sufficient, due to Fredholm theorem (see [274]),that ψi is satisfied the equation

4ψi =−4πqi

mic2

n∑k=1

qk

∫�1

(V,di)fkdV.

Furthermore, vector

Ci(r)di+ di× J(r) (7.5.17)

is a general solution of (7.5.16) with

J4=

4π

c

n∑k=1

qk

∫�1

V fkdV,

Ci—arbitrary function. Taking into account (7.5.12)–(7.5.14), it is easy to show thatfunctions ϕ,ψ satisfy the system

4ϕ =8παq

m

n∑k=1

qk

∫�1

fkdV, (7.5.18)

4ψ =−4πq

mc2

n∑k=1

qk

∫�1

(V,d)fkdV, (7.5.19)

with α4= a1, q

4= q1, m

4= m1, d

4= d1.

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Lemma 7.1. Vector di× J(r) is a potential and unique solution of (7.5.16) satisfyingcondition (7.5.15).

Proof. Sinceψ satisfies (7.5.19), then (7.5.17) is a general solution of (7.5.16). Due to(7.5.15), one can put Ci ≡ 0. Therefore, di× J—unique solution of (7.5.15), (7.5.16).We show that di× J—potential. In fact

rot [di× J]=−(di,∇)J+ d(∇,J),

where

(∇,J)≡ 0, (di,∇)J = (di,∇)rotB= rot(di,∇)B.

Due to (7.5.11),

(di,∇)B= (di,∇)

{λi

d2i

di− [di×∇ψi]mic

qid2i

}=

di

d2i

(∇λi,di)−mic

qid2i

×

× [di×∇(di,∇ψi)], (∇λi,di)= 0, (∇ψi,di)= 0.

Hence, rot [di× J]≡ 0, di× J =∇λi, and Lemma is proved.

Corollary 7.1.

∇λi(r)= [di× J(r)].

Lemma 7.2. Let b(x)= (b1(x),b2(x),b3(x)), x ∈ R3,

∂bi

∂xj=∂bj

∂xi, i, j= 1,2,3.

Then b(x)=∇λ(x), where

λ(x)=

1∫0

(b(τx),x)dτ +Const.

The proof is developed by straight calculation.

Corollary 7.2.

di

d2i

λi =d

d2

β + 1∫0

(d× J(τx),x

)dτ

, i= 1, . . . ,N, β −Const. (7.5.20)

The result follows from Lemma 7.2, Corollary 7.1, and (7.5.14).

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We are searching for the solutions (7.5.18), (7.5.19) satisfying orthogonality con-ditions (7.5.9), (7.5.10). Assuming d1i 6= 0, i= 1,2,3, we shall seek solutions in theform ϕ = ϕ(ξ,η), ψ = ψ(ξ,η)

ξ =

(y

d12−

z

d13

)+

d211

d211+ d2

12

(x

d11−

y

d12

),

η =|d1|d11d12

d13(d211+ d2

12)

(x

d11−

y

d12

), d1

4= (d11,d12,d13). (7.5.21)

Moreover, the problem is reduced to the study of nonlinear (semilinear) elliptic equa-tions

4ϕ = µ

n∑k=1

qk

∫�1

fkdV, (7.5.22)

4ψ = ν

n∑k=1

qk

∫�1

(V,d)fkdV, (7.5.23)

where

d4= d1, 4· =

∂2·

∂ξ2+∂2·

∂η2;

µ=8παq

mw(d); ν =−

4πq

mc2w(d); w(d)=

d2

d13(d211+ d2

12).

We note that every solution (7.5.22), (7.5.23), due to (7.5.21), satisfies orthogonalityconditions (7.5.9), (7.5.10). From preceeding it follows

Theorem 7.1. Let distribution function have the form (7.5.6). Then electromagneticfield {E,B} is defined by formulas

E(r)=m

2αq∇ϕ,

B(r)=d

d2

β +1∫

0

(d× J(τ r), r)dτ

− [d×∇ψ(r)]mc

qd2,

where r4= (ξ,η); β −Const; functions ϕ(r), ψ(r) satisfy system (7.5.22), (7.5.23).

Let us introduce a scalar and vector potentials U(r), A(r)

E(r)=−∇U(r), B(r)= rotA.

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Then, due to (7.5.7), (7.5.11), and (7.5.20), field {E,B} is defined via potentials {U,A}by formulas

U =−m

2αqϕ, A=

mc

qd2ψd+A1(r),

where (A1,d)= 0. Unknown potentials U,A can be defined in a subspace D of enoughsmooth functions on the set �⊂ R3 with a smooth boundary ∂� and moreover tosatisfy conditions

(∇U,d)= 0, (∇(A,d),d)= 0

and on the boundary

U|∂�2 = u0(r), (A,d)|∂�2 = u1(r). (7.5.24)

Corollary 7.3. Let distribution function be (7.5.6). Then the VM system (7.5.1)–(7.5.5)with boundary conditions (7.5.24) has a solution

fi = fi(−αi|V|2+ c1i+ liϕ

∗(r),diV + c2i+ kiψ∗(r)),

where li,ki satisfy (7.5.13) and (7.5.14),

E =m

2αq∇ϕ∗(r),

B=d

d2

β +1∫

0

(d× J∗(τ r),r)dτ

− [d×∇ψ∗(r)]mc

qd2,

J∗(r)=4π

c

n∑k=1

qk

∫�1

Vf dV.

Functions ϕ∗,ψ∗ belong to D and are defined from system (7.5.22), (7.5.23) withboundary conditions

ϕ|∂�2 =−2αq

mu0(r), (7.5.25)

ψ |∂�2 =q

mcu(r). (7.5.26)

7.5.2 Reduction of System (7.5.22), (7.5.23) to Single Equation

Lemma 7.3. If

f (V + d,r)= f (−V + d,r), d ∈ R3,

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then the following inequality holds

j= d · ρ, (7.5.27)

where j=∫�1

V f dV is vector of current density and ρ =∫�1

f dV—charge density.

Proof. Making changes of variables in integral∫�1

V f dV of the form Vi = ξi+ di (i=1,2,3), one obtains∫

Vi f (V,r)dV = J1+ J2+ J3,

where

J1 = di

∫�1

f (ξ + d,r)dξ,

J2+ J3 =

∞∫0

∞∫0

∞∫0

ξi f (ξi+ d,r)dξ +

0∫−∞

0∫−∞

0∫−∞

ξi f (ξi+ d,r)dξ.

It is easy to show that J3 =−J2 and (7.5.27) follows.

Taking into account Lemma 7.3, (7.5.22), (7.5.23) can be transformed to the form

4ϕ = µ

n∑i=1

qiAi, (7.5.28)

4ψ =νd2

2α

n∑i=1

kiqi

liAi, (7.5.29)

where Ai =∫�1

fidV , i= 1, . . . ,N.

Let (ξ,η) ∈�, where� is bounded domain in R2 with a smooth boundary ∂�. Weset a value of scalar potential on the boundary ∂�:

ϕ(ξ,η)|∂� = A(ξ,η). (7.5.30)

Consider when (7.5.28), (7.5.29) is reduced to one equation.

Case 1. li = ki, i= 1, . . . ,N.

Lemma 7.4. If li = ki and u∗ satisfies equation

4u= a(d,α)n∑

k=1

qiAi(γi+ liu) (7.5.31)

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with

γi = c1i+ c2i, i= 1, . . . ,N, u= ϕ+ψ

a(d,α)= 2πq4α2c2

− d2

mc2αw(d),

then system (7.5.28), (7.5.29) possesses a solution

ϕ =2(d,α)u∗+ϕ0,

ψ = (1−2(d,α))u∗−ϕ0,

where

2(d,α)= 4α2c2/(4α2c2− d2), 4α2c2

6= d2.

Knowing some solution u∗ of the equation (7.5.31) being solved under the condi-tions of Lemma 7.4 and the value of potential on the boundary ϕ|∂� = A(ξ,η), onefinds ϕ0 by means of solution of linear problem

4ϕ0 = 0,

ϕ0|� = A(ξ,η)−2u∗|∂�.(7.5.32)

Hence, in the first case, we transformed the problem to a solution of “solving” equa-tion (7.5.31) and the linear Dirichlet problem (7.5.32). This has the following result:

Theorem 7.2. Let ki = li, i= 1, . . . ,N. Then the VM system (7.5.1)–(7.5.5) withboundary condition (7.5.30) has a solution

fi = fi(−αi|V|2+Vdi+ γi+ liu

∗(ξ,η)),

E =m

2αq(2(d,α)∇u∗(ξ,η)+∇ϕ0),

B=d

d2

β +1∫

0

(d× J(τ r), r)dτ

−− [d× (∇(1−2(d,α))u∗(ξ,η)−ϕ0)]

mc

qd2.

u∗(ξ,η)—function satisfying “solving” equation (7.5.31); γi,βi—Const; r4= (ξ,η)

and ϕ0(ξ,η) is a harmonic function defined from linear problem (7.5.32).

Case 2. l2 = ·· · = ln4= l, k2 = ·· · = kn

4= k, l 6= k. We note that for N = 2, Cases 1

and 2 exhaust all possible connections between parameters li and ki. We constructsolution ϕ,ψ of (7.5.28), (7.5.29) satisfying condition

ϕ+ψ = lϕ+ kψ.

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Let functions fi4= fi(−αi|V|2+Vdi+ϕi+ψi) such that the following condition

holds.

(A). There are constants γi, i= 1, . . . ,N such that

2qA1(γ1+ u)+ τn∑

i=2

qiAi(γi+ u)= 0

for

2= 4α2c2(1− l)+ d2(k− 1), τ = 4α2c2(1− l)+ d2(k− 1)k

l.

We remark that the corresponding distribution function satisfies the condition ofLemma 7.3.

Lemma 7.5. Let l2 = l3 = ·· · = ln4= l, k2 = k3 = ·· · = kn

4= k, l 6= k. We assume that

condition (A) holds. Then (7.5.28), (7.5.29) possesses a solution

ϕ =k− 1

k− lu∗, ψ =

1− l

k− lu∗,

where u∗ satisfies equation

4u= εh

a(α, l)+ εb(d,k, l)A1(γ1+ u), (7.5.33)

ε =1

c2, h=

d2(k− l)28παq2

mw(d), a= 4α2(1− l)l, b= d2(k− 1)k.

Proof. By change ϕ = lu, ψ = ku system is reduced to (7.5.33), due to (A). Since

ϕ =k− 1

k− lu, ψ =

1− l

k− lu.

From Lemma 7.5 one obtains:

Theorem 7.3. Let α2q2/m2 = ·· · = αnqn/mn, k2 = ·· · = kn4= k. Let k /∈ {αnqn

mn,1} and

condition (A) holds. Then the VM system (7.5.1)–(7.5.5) with boundary condition(7.5.30) on scalar potential ϕ has a solution

fi = fi(−αi|V|2+Vdi+ γi+ u∗),

E =m(k− 1)

2αq(k− l)∇u∗,

B=d

d2

β +1∫

0

(d× J(τ r), r)dτ

− [d×∇u∗]cm(1− l)

qd2(k− l).

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Here u∗ satisfies (7.5.33) with condition

u∗|∂� =k− 1

k− l

m

2αqA(ξ,η), (7.5.34)

β, γi—constants, r4= (ξ,η).

The problem (7.5.33), (7.5.34) at ε→ 0 possesses solution u∗ = u0+O(ε), whereu0 is a harmonic function satisfying condition (7.5.34). Existence of another solu-tions for equations (7.5.33), (7.5.34) can be proved using parameter continuation andbranching theory methods.

7.6 Existence of Solutions for the Boundary ValueProblem (7.5.28)–(7.5.30)

We realize the form of distribution function. Let

fi = exp(−αi|V|2+Vdi+ γi+ liϕ+ kiψ). (7.6.1)

Distributions (7.6.1) have meaning in applications. Substituting (7.6.1) into (7.5.28)and (7.5.29), taking into account (7.5.12)–(7.5.14) and (7.5.27), we come to the system

4ϕ = µ

n∑k=1

qi

(π

ai

)3/2

exp

(γi+

d2i

4αi

)exp(liϕ+ kiψ),

4ψ =d2ν

2α

n∑i=1

qi

(π

αi

)3/2

exp

(γi+

d2i

4αi

)exp(liϕ+ kiψ)

ki

li.

(7.6.2)

Introducing normalization condition∫�

∫R3

fidVdx= 1, i= 1, . . . ,N; � ∈ R2; x

4= (ξ,η),

we transform (7.6.2) to the form

4ϕ = µ

n∑i=1

qiexp(liϕ+ kiψ)

∫�

exp(liϕ+ kiψ)dx

−1

,

4ψ =d2ν

2α

n∑i=1

qiki

liexp(liϕ+ kiψ)

∫�

exp(liϕ+ kiψ)dx

−1

.

(7.6.3)

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Consider when it is not possible to transform (7.6.3) to one equation. Without loss

of generality, we can consider that l2 6= k2; q4= q1. Let q1 < 0, qi > 0, i= 2, . . . ,N.

Introducing new variables

u1 = ϕ+ψ, ui =−(liϕ+ kiψ), i= 2, . . . ,N.

And using them with boundary conditions (7.5.25)–(7.5.26), one obtains system

−4ui =

n∑j=1

CijAj, i= 1, . . . ,N, (7.6.4)

where

A1 = eu1

∫�

eu1 dx

−1

, Aj = e−uj

∫�

e−ujdx

−1

, j= 2, . . . ,N,

Cij =8π

w(d1)·

ai

mi|qi|qj

(1−

1

2d21c2

ZiZj

), Zi =

(d1,di)

αi,

ui = u0i, x ∈ ∂�, i= 1, . . . ,N.(7.6.5)

It is easy to check that (7.6.3) and (7.6.4) are equivalent in the sense that solutions(7.6.4) define completely solutions of (7.6.3). In fact ϕ,ψ are defined via u1,u2,because l2,k2 and ui are linear dependent for i= 3, . . . ,N.

Here we assume that u0i ∈ C2+α , ∂� ∈ C2+α , α ∈ (0,1). We give auxiliary results.

Lemma 7.6. Let

n∑j=1

Cij > 0,

n∑j=1

Cij < 0

.Then

Fi(u)=n∑

j=1

CijAj(u)≥ 0, ui ≥min∂�

u0i,Fi(u)=n∑

j=1

CijAj(u)≤ 0, ui ≤max∂�

u0i

.Proof. It is easy to see that

∫�

Fi(u)dx=∑n

j=1 Cij > 0. Moreover, the set �+ = {x ∈

� : Fi(u(x)) > 0} is nonempty. We denote connected components in �−, i.e., maxi-mum (by inclusion) connected subspace �− = {x ∈� : Fi(u(x)) < 0}, and we show

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that �− = ∅. Hence, on the one hand, Fi(u(x))= 0, where x ∈ ∂�, and, on the otherhand,

−4ui(x)= Fi(u(x)) < 0, x ∈�.

Thus, ui is bounded in �, and it reaches it’s maximum on ∂�= �\�, i.e.,maxx∈� u(x)= u(x0), x0 ∈ ∂�. However, since function Fi(u) decreases for fixed(∫�

e−uj dx)−1, then one obtains Fi(u(x)) > Fi(u(x0))= 0, x ∈ � that contradicts def-

inition of the set �−. By analogy, case∑n

j=1 Cij < 0 is considered (see [165]).

Lemma 7.7 (Gogny, Lions [120]). Let

max�(u− v)(x)= (u− v)(x0) > 0.

Then

eu(x0)

∫�

eu(x)dx

−1

> ev(x0)

∫�

ev(x)dx

−1

,

e−u(x0)

∫�

e−u(x)dx

−1

< e−v(x0)

(∫�

e−v(x)dx

)−1

.

We define the vector-function v(x), w(x) ∈ C2(�)n ∩C1(�)n as a lower and anupper solution of (7.6.4), (7.6.5) in the following sense

−4vi ≤

n∑j=2

Cije−wj∫

�e−vjdx

+Ci1ew1∫

�ev1 dx

≤ Fi(v), x ∈�,

−4wi ≥

n∑j=2

Cije−vj∫

�e−wj dx

+Ci1ev1∫

�ew1 dx

≥ Fi(w), x ∈�,

(7.6.6)

vi ≤ u0i, wi ≥ u0i, x ∈ ∂� (7.6.7)

with v= (v1, . . . ,vn)′, w= (w1, . . . ,wn)

′.It is easy to show that Aj(u) is invariant under translation on the constant vector,

therefore, one can change on (7.6.7)

vi ≤ 0, wi ≥ 0, x ∈ ∂�. (7.6.8)

Theorem 7.4. Let there exist a lower vi(x) ∈ C2(�)∩C1(�) and an upper wi(x) ∈C2(�)∩C1(�) solution satisfying inequalities (7.6.6), (7.6.8), such that vi(x)≤ wi(x)in �. Let u0i ∈ C2+α(∂�). Then (7.6.4), (7.6.5) has a unique classical solution ui(x) ∈C2+α(�) and, moreover, vi(x)≤ ui(x)≤ wi(x) in �, i= 1, . . . ,N.

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Proof. Let functions zi(x) ∈ C(�) be given, vi ≤ zi ≤ wi. We define operatorT : C(�)n→ C(�)n by formulas u= Tz, z(x) ∈ ∈ C(�)n, where u= (u1, . . . ,un)

′ isa unique solution of the problem

−4ui =

n∑j=1

CijAj(p(z))+ q(zi)4= Fi(z), ui = u0i, x ∈ ∂�, (7.6.9)

where p(z)=max{v,min{z,w}},

q(zi)=

wi−zi

1+z2i, zi ≥ wi,

0, vi ≤ zi ≤ wi,vi−zi

1+z2i, vi ≤ zi.

It is evident that function F(z) is continuous and bounded. Then, due to smoothnessof ∂� and boundary conditions, (7.6.9) is only solvable in C1+α(�)n, i.e., u(x) ∈C1+α(�)n. Here we used Theorem 8.34 from [110]. Due to compactness of embeddingC1+α(�)⊂ C(�) and continuity of F(z), it follows that operator T is a completelycontinuous (compact) operator. Then by Schauder theorem (see [146]), operator T pos-sesses a fixed point u= Tu with u ∈ C(�)n. On the other hand, since u ∈ C1+α(�)n,then F(u) ∈ Cα(�)n and from classical theory follows that u ∈ C2+α(�)n.

Next we show that vi ≤ ui ≤ wi. We suppose that there exist a number k ∈{1, . . . ,N} and the point x0 ∈ � such that

(vk− uk)(x0)=max�(vk− uk)= ε > 0.

Evidently, x0, due to (7.6.7), can not belong to the boundary ∂�. Then due to maxi-mum principle, one has contradiction

0≤−4(vk− uk)(x0)≤ Ck1ew1(x0)∫�

ev1dx+

n∑j=2

Ckje−wj(x0)∫�

e−vjdx−

−Ck1ep(u1)(x0)∫�

ep(u1)dx−

n∑j=2

Ckje−p(uj)(x0)∫�

e−p(uj)dx+(uk− vk)(x0)

1+ u2k(x0)

< 0.

Thus, vi ≤ ui. By analogy, the proof of inequality ui ≤ wi is given.We assume that there exists a number l ∈ {1,2, . . . ,N} and the point y0 ∈ � such

that there are two solutions u1, u2 of (7.6.4), (7.6.5), u1i ≡ u2

i , i 6= l, u1l (y0) > u2

l (y0).Using Lemma 7.7, we come again to contradiction: 0≤−4(u1

l − u2l )(y0) < 0, which

proves uniqueness.

We construct upper and a lower solutions of (7.6.4), (7.6.5). Let∑n

j=1 Cij > 0,i= 1, . . . ,N. Then from Lemma 7.6 it follows ui ≥ 0. At first, we construct an upper

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solution of the form: vi ≡ 0,

−4wi =

n∑j=2

Cij∫�

e−wj dx−|Ci1|∫�

ew1 dx, (7.6.10)

wi|∂� =maxi,∂�

u0i ≡ w0 (7.6.11)

with x= (ξ,η) ∈�⊂ R2. From (7.6.6) follows that wi must be satisfied inequalities

n∑j=2

Cije−wj − |Ci1|e

w1 ≥ 0, i= 1, . . . ,N. (7.6.12)

Consider auxiliary problem

−4g= 1, g|∂� = w0.

We assume that domain � is contained in a strip 0< x1 < r, and one introduces thefunction q(x)= w0+ er

− ex1 . It is easy to show that

4(q− g)=−ex1 + 1< 0 in �,

q− g= er− ex1 ≥ 0 on ∂�. Therefore, according to maximum principle (see [110])

q− g≥ 0, if x ∈ � and

w0 ≤ g(x)≤ w0+ er− 1

4=M. (7.6.13)

We denote write part in (7.6.10) by zi = const≥ 0. Then from (7.6.10) and (7.6.13) weobtain wi ≤Mzi, wi = zig(x), and (7.6.10), (7.6.11) is equivalent the following finite-dimensional algebraic system

zi =

n∑j=2

Cij∫�

e−zjgdx−|Ci1|∫

�

ez1gdx4= Li(z).

Let us introduce the norm |z| =max1≤i≤N |zi|. Then, due to (7.6.13), we obtain thefollowing chain of inequalities

|L(z)| ≤ max1≤i≤n

∣∣∣∣ n∑j=2

Cij∫�

e−zjgdx−|Ci1|∫�

ez1gdx

∣∣∣∣≤≤

1

|�|max

1≤i≤N

n∑

j=2

CijeMzj − |Ci1|e

−Mz1

≤≤

1

|�|max

1≤i≤N

n∑

j=2

CijeM|z|− |Ci1|e

−M|z|

, (7.6.14)

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where |�| =mes �, �⊂ R2.

Lemma 7.8. Let∑n

j=1 Cij > 0. Introduce notations

n∑j=2

Cij4= ai, |Ci1| = bi, min

1≤i≤N

ai

bi= α2 > 1.

Let the inequalities

αai−1

αbi ≤|�|

Mlnα, i= 1, . . . ,N (7.6.15)

hold. Then equation Lz= z has a solution zi ≤1M ln α and functions vi ≡ 0, wi = zig(x)

are a lower and an upper solutions of the problem (7.6.4), (7.6.5).

Proof. Let |z| = R. From (7.6.12) follows

aie−MR− bie

MR≥ 0

with R≤ 1M lnα. Substituting a maximum value R= 1

M lnα in (7.6.14), it is easy tocheck that (7.6.15) gives estimation |L(z)| ≤ |z| and existence of the fixed point Lz= zfollows from Brayer theorem (see [146]).

Let now∑n

j=1 Cij ≤ 0, i= 1, . . . ,N. By analogy with preceeding we obtain thefollowing result.

Lemma 7.9. Let∑n

j=1 Cij < 0, β2=min1≤i≤N(bi/ai) > 1 and the inequalities

bi

β−βai ≤

|�|

Mlnβ, i= 1, . . . ,N (7.6.16)

hold. Then functions vi =−zig(x), wi ≡ 0 are a lower and an upper solutions of(7.6.4), (7.6.5) with zi =−Li(−z).

It follows from Theorem 7.4 and smoothness of the function Fi(u) under the fixedfunctional coefficients (

∫�

e−uj dx)−1 that there exists a constant M(v,w) > 0 such that

∂∂uj

Fi ≥−M with i, j= 1, . . . ,N. Moreover the mapping G : C(�)n→ C(�)n definedby formulas Giu= Fi+Mui will be monotonic, increasing in ui because of mono-tonicity of coefficients. We set operator T1 : z= T1z,

−4zi+Mzi = Giu> 0, zi|∂� = u0i. (7.6.17)

Due to maximum principle, zi > 0 (u0i > 0). Thus, operator T1 is positive and mono-tonic. Moreover, T1 is completely continuous, being proven in the same way that foroperator T also. It is evident, v≤ T1v, T1w≤ w. We note that a cone of nonnegativefunctions is normal in C(�). Therefore, due to uniqueness Theorem (7.4), we canapply the classical theory of monotone operator (see [160]) for problem (7.6.17) andobtain the following result:

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Theorem 7.5. Operator T1 has a unique fixed point u= T1u, vi ≤ ui ≤ wi, wherefor any y0 : vi ≤ y0i ≤ wi, successive approximations yn+1 = T1yn are uniformly con-verged to u.

Corollary 7.4. We define successive approximations by

u0i = 0,

−4un+1i +Mun+1

i = Fi(un)+Mun

i ,

un+1i |∂�=u0i , i= 1,2, n= 0,1, . . . ;

unk =

qkαk

mk(z2− z1)

[m1

|q1|α1(z2− zk)u

n1+

m2

q2α2(zk− z1)u

n2

], k = 3, . . . ,n.

Then {uni }, i= 1, . . . ,n are monotone and uniformly converged to solution (7.6.4),

(7.6.5).

Remark 7.1. In the case n= 1, boundary value problem (7.6.4), (7.6.5) was consid-ered in Gogny, Lions [120] and Krzywicki, Nadzieja [165].

7.7 Existence of Solution for Nonlocal BoundaryValue Problem

Here we consider the problems (7.5.28), (7.5.29), (7.5.25), (7.5.26). Assume plasma indomain �⊂ R2 with a smooth boundary ∂� ∈ C1 consisting of N kinds of chargedparticles. It is assumed that particles interact among themselves only by means ofowns charges q1, . . . ,qn ∈ R\{0}. Every particle of i-th kind is described by distributionfunction fi = fi(x,v, t)≥ 0, where t ≥ 0—time, x ∈�—position and v ∈ R3—velocity.Plasma motion is described by the classical VM system (7.5.1)–(7.5.5) with boundaryconditions (7.5.24). We impose reflection condition (7.3.1) for distribution function.

In this section we study stationary solutions ( f1, . . . , fn,E,B) of the VM system ofspecial form

fi = fi(−αiv2+ c1i+ liϕ(x),vdi+ c2i+ kiψ(x)), (7.7.1)

E(x)=m

2αq∇ϕ, (7.7.2)

B(x)=−cm

qd2(d×∇ψ), (7.7.3)

where functions fi : R2→ [0,∞) and parameters d ∈ R3

\{0}, αi > 0, c1i, c2i, li,ki (seeformulas of connection (7.5.12)–(7.5.14)) are given, and functions ϕ,ψ have to bedefined. Earlier, using the lower-upper solutions method, existence theorem of clas-sical solutions of boundary value problem (7.5.28)–(7.5.30) is proved for distributionfunction fi = eϕ+ψ . Under proof existence Theorem 7.4, we essentially apply mono-tonic property of the right parts of (7.6.4). In general case of distribution function

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(7.7.1), system (7.5.28), (7.5.29) does not possesses good monotonic properties and,therefore, we can not apply techniques of lower and upper solutions for nonlinearelliptic system in a cone developed by Amann [3]. Therefore, we show existence ofsolutions of the boundary value problem (7.5.28), (7.5.29), (7.5.25), (7.5.26) by themethod of lower-upper solutions without monotonic conditions. We note that approach(7.7.1)–(7.7.3) is connected with papers of P. Degond [86] and J. Batt, K. Fabian [27].In these papers they are introduced integrals f (E), F(x,v) and P(x,v) of the Vlasovequation and solutions of the VM system for distribution function (i= 1—particles ofsingle kind) of the form f (E), f (E,F) or f (E,F,P) are considered. The case of distri-bution function of f (E,P) and particles of various kinds (species i= 1, . . . ,N) in thesepapers are not considered.

Thus, we consider the boundary value problem (7.5.28), (7.5.29), (7.5.25), (7.5.26).Let q< 0 (electrons), qi > 0 (positive ions), i= 2, . . . ,N. Then (7.5.28), (7.5.29) takesthe form

4ϕ =8παq

mw(d)

(qA1−

n∑i=2

|qi|Ai

)= h1, (7.7.4)

−4ψ =4πq

mc2w(d)

d2

2α

(qA1−

n∑i=2

ki

li|qi|Ai

)= h2, (7.7.5)

where Ai =∫�

fidv, i= 1, . . . ,N, and fi is ansatz (7.7.1).

Remark 7.2. In case ki = li, system (7.7.4), (7.7.5) is transformed to one equation,and we may use Theorem 7.4.

Theorem 7.6 (McKenna-Walter [194]). Let�⊂ Rn—bounded domain with boundary∂� ∈ C2,µ for some µ ∈ [0,1]. Let h : �×Rn

→ Rn satisfy the following smoothnessconditions: ∀r > 0 there exist Cr > 0 such that ∀x,x1,x2 ∈ � and ∀y,y1,y2 ≤ r:

I. There hold the inequalities

|h(x1,y)− h(x2,y)| ≤ Cr|x1− x2|µ,

|h(x,y1)− h(x,y2)| ≤ Cr|y1− y2|;

II. There exists ordered couple (v,w) of lower v and upper w solutions, i.e., v,w ∈C2(�)n

⋂C1(�)n, v≤ w in �, v≤ 0≤ w on ∂�,

∀x ∈� : ∀z ∈ Rn, v(x)≤ z≤ w(x), zk = vk(x) :4vk(x)≥ hk(x,z)

and

∀x ∈� : ∀z ∈ Rn, v(x)≤ z≤ w(x) : zk = wk :4wk(x)≤ hk(x,z)

for all k ∈ {1, . . . ,N} (Here the vector inequality v(x)≤ z≤ w(x) means a component wisecomparison).

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Then there is a solution u ∈ C2,µ(�)n of the problem

4u= h(·,u(·)) in �

u= 0 ∂�

such that v≤ u≤ w in �.

Because the right parts in (7.7.4), (7.7.5) are nonlocal, we give sufficient conditionsto function fi to make it possible to apply a McKenna-Walter theorem.

Lemma 7.10. Let α > 0 and f : R2→ [0,∞) satisfy the following conditions:

1. f ∈ C1(R2);

2. f and f ′ are bounded there exists R0 ∈ R such that supp(f )⊂ [R0,∞)×R.Then function h

α,f : R2→ R2, given via

hα,f (u)=

4πq

mw(d)

∫R3

(2αq−

1c2

kili

)f (−αv2

+ u1,vd+ u2)dv

is continuously differentiable, and there are R,C1,C2 such that(0

−C2(u1+R)2+

)≤ h

α,f (u)≤

(C1(u1+R)3/2+C2(u1+R)2+

)

for any function u ∈ R2.

Proof. Passing on to a spherical system of coordinates

v1 = ρ sin2cosϕ, v2 = ρ sin2sinϕ, v3 = ρ cos2,

we obtain

h1α, f(u)=

8παq2

mc2w(d)

∫R3

f (−αv2+ u1,vd+ u2)dv=

= P(q,α,d,m)

∞∫0

π∫0

2π∫0

f (−αϕ2+ u1,ϕk(ρ,2)+ u2)sin(2)ϕ2dρd2dϕ =

=P(q,α,d,m)

α2

u1∫−∞

π∫0

2π∫0

f (s,α−1k(ρ,2)√(u1− s)+ u2)×

× sin(2)(u1− s)√(u1− s)dρd2ds=

=P(q,α,d,m)

α2

u1∫−∞

K1(s,u1− s,u2)(u1− s)√(u1− s)ds,

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where

k(ρ,2)= d1 cos(ρ)sin(2)+ d2 sin(ρ)sin(2)+ d3 cos(2)

and

K1(s, t,ϕ)=

π∫0

2π∫0

f (s,α−1k(ρ,2)√

t+ϕ)sin(2)dρd2.

Similar expressions are satisfied for h2α,f

and K2(s, t,ϕ). Due to condition (2) kernels

K1,K2 are bounded and applying Lebesgue theorem on dominant convergence, it iseasy to prove that h

α,f ∈ C1(R2)2.

Theorem 7.7. Let �⊂ R2—two-dimensional domain with boundary ∂� ∈

∈ C2,µ, µ ∈ [0,1]. Let f1, . . . , fn : R2→ [0,∞) satisfy conditions (1), (2) of

Lemma 7.10. Then the problem (7.5.28), (7.5.29), (7.5.25), (7.5.26) possesses a smoothsolution ϕ ∈ C2(�), ψ ∈ C2(�). Moreover distribution function fn ∈ C1(�×R3)

generates the classical stationary solution ( f1, . . . , fn,E,B) of the VM system of theform (7.7.1)–(7.7.3) in �.

Proof. Consider the system (7.7.4), (7.7.5). The right parts of these expressions mayhave different signs, depending on relations

A1. qA1−∑n

i=2 |qi|Ai = G(q,A) > 0. Hence,

qA1 >

n∑i=2

|qi|Ai >

n∑i=2

T−|qi|Ai.

A2. qA1−∑n

i=2 |qi|Ai = G1(q,A) < 0. Hence,

qA1 <

n∑i=2

|qi|Ai <

n∑i=2

T+|qi|Ai.

Here

T− =min

{ki

li

}=min

{(di,d)α

d2i αi

},

T+ =max

{ki

li

}=max

{(di,d)α

d2i αi

}.

It follows from Lemma 7.10 and conditions (A1), (A2) that right parts h1,h2 of (7.7.4),(7.7.5) satisfy smoothness conditions of McKenna-Walter theorem, and there are R>0 and matrix (2×N) with positive components such that(

−∑

G1<0 c1i|G1|(liu1+R)2+−∑

G>0 c2i|G|(liu1+R)2+

)≤ h(u)≤

( ∑G>0 c1i|G|(liu1+R)3/2+∑G1<0 c2i|G1|(liu1+R)2+

)

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for all u ∈ R2. Now we pass to construction of lower-upper solutions (v,w) of (7.7.4),(7.7.5), (7.6.4), (7.6.5). Introduce the following notations

l+ =min{|li| | li > 0}, l− =min{|li| | li < 0}

and

l=min(l+, l−).

We define a lower and an upper solution in �

v=

−εl+

4−1∑n

i=1 c2i|G|

(1+ |li|l

)2

R2

and

w=

εl−

−4−1∑n

i=1 c2i|G|

(1+ |li|l

)2

R2

and on the boundary

vi ≤ u0i, wi ≥ u10i, x ∈ ∂�

with v= (v1,v2)′,w= (w1,w2)

′. Assuming that the right parts h1(·),h2(·) of (7.7.4),(7.7.5) are invariant under the transition on the constant vector, we can change lastconditions on the following ones

vi ≤ 0, wi ≥ 0, x ∈ ∂�.

Moreover operator 4−1 is defined with respect to zero boundary conditions and v≤0≤ w in �.

Due to the above given estimation for hf and conditions (A1), (A2), we obtain

4v1 = 0≥ h1f (v1,z2), z2 ∈ R,

4w1 = 0≤ h1f (w1,z2), z2 ∈ R,

4v2 ≥

n∑i=1

c2i|G|(liz1+R)2+ ≥ h2f (z1,v2), z1 ∈ [v1,w1]

and

4w2 ≤−

n∑i=1

c2i|G|(liz1+R)2+ ≤ h2f (z1,w2), z1 ∈ [v1,w1].

Thus, existence of solutions U ∈ C2,µ(�), U = (ϕ,ψ)′ of (2.54), (2.55) (respectively(2.40, (7.5.29)) (7.5.25), (7.5.26) follows from McKenna-Walter theorem.

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Remark 7.3. Existence of stationary solutions for the relativistic VM system has beenproven in the dissertation of P. Braasch [55] with using RSS [183] ansatz.

7.8 Nonstationary Solutions of the Vlasov-Maxwell System

7.8.1 Reduction of the Vlasov-Maxwell System to NonlinearWave Equation

Let us consider the nonstationary VM system (7.1.2)–(7.1.6) for N-component distri-bution function with additional condition

n∑i=1

q2i

mi

∫R3

{E+

1

c[V ×B]

}·∇V fidV = 0. (7.8.1)

We shall seek distribution functions of the form

fi = fi(−αi|V|2+Vdi+Fi(r, t)), di ∈ R3, αi ∈ [0,∞) (7.8.2)

and corresponding fields E(r, t),B(r, t) satisfying equations (7.1.2)–(7.1.6), (7.8.1). Iffunctions Fi(r, t), vectors di and vector-functions E,B are connected among them-selves by relations

∂Fi

∂t+

qi

mi(E,di)= 0, (7.8.3)

∇Fi−2αiqi

miE+

qi

mic[B× di]= 0, i= 1, . . . ,N, (7.8.4)

then functions (7.8.2) satisfy (7.1.2) and one has the following equations

∂Fi

∂t+

1

2αi(∇Fi,di)= 0,

∂fi∂t+

1

2αi(∇fi,di)= 0.

(7.8.5)

Introducing auxiliary vectors Ki = (Kix(r, t),Kiy(r, t),Kiz(r, t)), we transform (7.8.4)to the system

∇Fi−2αiqi

miE = Ki, (7.8.6)

qi

mic[B× di]=−Ki. (7.8.7)

We note that equation (7.8.7) is solvable with respect to vector B, if

(Ki,di)= 0. (7.8.8)

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We define functions Fi(r, t) and vectors Ki(r, t) in the form

Fi = λi+ liU(r, t),

Ki = kiK(r, t),

where λi,ki, li—constants, l1 = k1 = 1. Then from (7.8.6) and (7.8.7) follows that

E(r, t)=mi

2αiqi(li∇U− kiK),

B(r, t)=γ

d2i

di+ [K× di]kimic

qid2i

,

where γi(r, t)= (B,di) are remained arbitrary functions. Let

li = ki =m1

α1q1

αiqi

mi,

αid1 = α1di, αiγ1 = α1γi, i= 1, . . . ,N.

Then

E(r, t)=m

2αq(∇U−K), (7.8.9)

B(r, t)=γ

d2d+ [K× d]

mc

qd2, (7.8.10)

where the following notations are introduced

m4= m1, α

4= α1, d

4= d1, γ

4= γ1.

Moreover, K ⊥ d. Due to (7.8.3), (7.8.8), function U(r, t) satisfies linear equation

2α∂U

∂t+ (∇U,d)= 0. (7.8.11)

Having defined U,K such that the Maxwell equations (7.1.2)–(7.1.5) to be satisfiedfor distribution function

fi = fi(−αi|V|2+Vdi+ λi+ liU(r, t)), (7.8.12)

we can find unknown functions fi,E,B using (7.8.9), (7.8.10) and (7.8.12).

Lemma 7.11. Densities of charge ρ and current j defined by formulas

ρ(r, t)= 4π∫R3

n∑i=1

qi fidV, j(r, t)= 4π∫R3

n∑i=1

qiVfidV,

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are connected among themselves by the following relation

j=1

2αdρ+ rotQ(r)+∇ϕ0(r), 4ϕ0(r)= 0. (7.8.13)

An equality (7.8.13) follows directly from continuity equation

∂ρ

∂t+∇ × j= 0

and

∂ρ

∂t+

1

2α(d,∇ρ)= 0,

which is corollary of (7.8.5).Substituting (7.8.9), (7.8.10) into (7.1.3), (7.1.5), one obtains

4U = divK+8παq

m

n∑i=1

qi

∫R3

fidV, (7.8.14)

(d,∇γ )+mc

q(d, rotK)= 0. (7.8.15)

Due to Lemma 7.11 and taking into account rotQ(r)+∇ϕ0= 0 (that always can be

assured by calibrating),∫R3

V fidV =d

2α

∫R3

fidV.

Thus, after substitution (7.8.9), (7.8.10) into (7.1.2), we obtain the relation

∇γ × d =md2

2αcq

∂

∂t(∇U−K)+

2πd2

αcd

n∑i=1

qi

∫R3

fidV −mc

qrot [K× d]. (7.8.16)

Having used the Fredholm alternative, we set the function U(r, t), and from conditionthat its solution ∇γ is a gradient of function γ (r, t), we find K(r, t) as function of U.Thus, from solvability condition of (7.8.16) with respect to (7.6.16), one obtains

∂2U

∂t2=

2πqd2

αm

n∑i=1

qi

∫R3

fidV + c2divK.

Due to (7.8.14), this equility is transformed into

∂2U

∂t2= c24U+

2πq

αm(d2− 4α2c2)

n∑i=1

qi

∫R3

fidV. (7.8.17)

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Below we apply (7.8.17) for solvability (7.8.16). If function U satisfies (7.8.17), then(7.8.16) is satisfied and, moreover,

∇γ =ν

d2d+

[d×

{−

mc

qrot [K× d]+

md2

2αcq

∂

∂t(∇U−K)

}]1

d24= F, (7.8.18)

where ν(r, t)= (∇γ,d) is kept arbitrary. It follows from (7.8.18) that vector fieldF(r, t)must be irrotational. Since U satisfies (7.8.11), we define K in a class of vectorssatisfying condition

2α∂K

∂t+ (d · ∇)K = 0. (7.8.19)

Then d× rot [K× d]=−2α[d× ∂K/∂t] and (7.8.18) transforms

∇γ =ν

d2d+

[d×

{(4α2c2

− d2)∂K

∂t+ d2 ∂

∂t∇U

}]m

2αcqd2.

Up to arbitrary function b(U) and arbitrary vector-function a(r), one can put

K(r, t)=d2

d2− 4α2c2(∇U+ b(U)d+ a(r)). (7.8.20)

Then

∇γ =ν

d2. (7.8.21)

If

b(U)=−1

d2(∇U,d), a(r)=∇ϕ0(r),

where ∇ϕ ⊥ d, then (7.8.20) satisfies (7.8.19). Proof is developed by direct substitu-tion of (7.8.20) into (7.8.19) with account of (7.8.11). Thus, vector

K(r, t)=d2

d2− 4α2c2

{∇U−

1

d2(∇U,d)d+∇ϕ0(r)

}, (7.8.22)

where ∇ϕ ⊥ d satisfies condition (7.8.19). Moreover, it is evident that K ⊥ d. If

4ϕ0(r)= 0,

then for any U(r, t) satisfying (7.8.17) vector-function (7.8.22) satisfies (7.8.14) thatcan be showed by substitution (7.8.22) into (7.8.14). We show that in (7.8.21) ν ≡ 0.In fact,

(d, rot((∇U,d))= (d,∇(∇U,d)× d)≡ 0

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for arbitrary U, (d, rotK)= 0 and, due to (7.8.15), d ⊥∇γ . But then from (7.8.21),ν ≡ 0. Therefore, ∇γ = 0, and γ is a constant.

It remains to show that functions (7.8.9), (7.8.10), where U(r, t) satisfies (7.8.17)and K(r, t) are expressed via U and ϕ0 by formula (7.8.22), satisfy (7.1.4). From sub-stitution (7.8.9) and (7.8.10) in (7.1.4), we obtain the chain of equalities

m

q

{1

d2

[∂K

∂t× d

]−

1

2αrotK

}=

=m

q(d2− 4α2c2)

{∂

∂t[∇U× d]+

1

2αrot((∇U,d)d)

}=

=m

q(d2− 4α2c2)

[∇

(∂U

∂t+

1

2αrot(∇U,d)

)×d

]= 0.

Remark 7.4. If (7.8.13) holds, then functions γ 6= Const, ∇γ = d× rotQ.

Hence, it follows.

Theorem 7.8. Let fi(S)—an arbitrary differentiable functions, moreover∫R3

fi(−|V|2+T)dV <∞, T ∈ (−∞,+∞), αi ∈ [0,∞), di ∈ R3,

αid = αdi, α4= α1, d

4= d1,

then every solution U(r, t) of hyperbolic equation (7.8.17) with condition (7.8.11) cor-responds solution of the system (7.1.1)–(7.1.5) of the form

fi = fi(−αi|V|2+Vdi+ λi+ liU(r, t)), (7.8.23)

B=γ

d2d+

mc

q(d2− 4α2c2)[∇(U+ϕ0(r))× d],

E =m

2αq(4α2c2− d2){∇(4α2c2U+ d2ϕ0(r)− (∇U,d)d)},

(7.8.24)

where ϕ0(r)—arbitrary function satisfying 4ϕ0 = 0, ∇ϕ0 ⊥ d.

Corollary 7.5. In the stationary case, (7.8.17) is transformed to the form

4U(r)=2πq

αmc2(4α2c2

− d2)

n∑i=1

qi

∫R3

fidV (7.8.25)

with condition

(∇U,d)= 0. (7.8.26)

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Remark 7.5. If

fi = es, S=−αi|V|2+Vdi+ λi+ liU, li =

αimqi

αmiq,

then ∫R3

fidV = (π

αi)3/2exp{d2

i /4αi+ λi+ liU}.

In that case, “solving” equation (7.8.17) is written

∂2U

∂t2= c24U+

2πq

αm(d2− 4α2c2)π3/2

n∑i=1

qi(αi)−3/2exp{d2

i /4αi+ λi+ liU}.

As seen in paper [184], for N = 2 (two-component system), this equation is trans-formed to:

∂2U

∂t2= c24U+ λb(eU

− elU), l ∈ R−, λ ∈ R+,

b=2πq2

αm

(π

α

)3/2

(d2− 4α2c2)ed2/4α.

(7.8.27)

Due to l=−1, (7.8.27) is a wave sh-Gordon equation

∂2U

∂t2= c24U+ 2λbsinhU. (7.8.28)

Remark 7.6. Due to conditions of Theorem 7.8, a scalar 8 and vector A potentialsare defined by formulas

8=m

2αq(d2− 4α2c2){4α2c2U(r, t)+ d2ϕ0},

A=mc

q(d2− 4α2c2)d{U(r, t)+ϕ0}+42(r),

(7.8.29)

where

42(r)=γ

d2(d2z,d3x,d1y)′+∇p(r), d

4= (d1,d2,d3) (7.8.30)

and p(r) is arbitrary harmonic function. Since function U(r, t) satisfies (7.8.11), thenpotentials 8,A are connected among themselves by Lorentz calibration.

1

c

∂8

∂t+ divA= 0.

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For analysis (7.8.28), we direct a constant vector d ∈ R3 along axis Z, i.e., we

assume that d4= d1(0,0,dz). Moreover, solution U(x,y,z, t) for (7.8.11) has the form

U = U(x,y,z−d

2αt). (7.8.31)

Solution (7.8.31) describes the wave spreading velocity running in positive directionalong axis Z with a constant velocity d/2α, where d/2α < c. By substitution ξ =z− (d/2α)t, we reduce (7.8.28) to

∂2U

∂x2+∂2U

∂y2+(4α2c2

− d2)

4α2c2

∂2U

∂ξ2= 2λp sinhU, (7.8.32)

where

p4=

2πq2

αmc2

(π

α

)3/2

(4α2c2− d2)exp(d2/4α) > 0; λ ∈ R+.

Moreover, introducing a new variable η = (4α2c2/(4α2c2− d2))1/2ξ , we transform

(7.8.32)

∂2U

∂x2+∂2U

∂y2+∂2U

∂η2= 2λp sinhU, U

4= U(x,y,η). (7.8.33)

Using formulas (2.69), it is easy to reconstruct some solutions of (7.8.33) by Hirotamethod [134].

7.8.2 Existence of Nonstationary Solutions of the Vlasov-MaxwellSystem in the Bounded Domain

Here we consider the classical solutions ( f1, . . . , fn,E,B) of the VM system of specialform (7.8.23)–(7.8.24), which we write in the following form:

fi(x,v, t)= fi(−αiv2+ vdi+ liU(x, t)), (7.8.34)

E(x, t)=m

2αq(4α2c2− d2)

(4α2c2

∇U(x, t)+ ∂tU(x, t)d), (7.8.35)

B(x, t)=−mc

q(4α2c2− d2)∇U(x, t)× d, (7.8.36)

where functions fi : R→ [0,∞) and vector d ∈ R3\{0} are given, and function U :

[0,∞)× �→ R has to be defined. Assuming that ∂� ∈ C1, we add the VM systemby the boundary conditions for electromagnetic field

E(x, t)× n�(x)= 0, B(x, t)n�(x)= 0, t ≥ 0, x ∈ ∂�, (7.8.37)

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and specular reflection condition for distribution function on the boundary

fi(t,x,v)= fi(t,x,v− 2(vn�(x))n�(x)), t ≥ 0, x ∈ ∂�, v ∈ R3, (7.8.38)

where n�—unit vector of normal to ∂�.To prove existence of classical solutions of (7.1.2)–(7.1.6), (7.8.34)–(7.8.38), we

apply the method of lower-upper solutions developed for nonlinear elliptic systems.In contrast to stationary problem, nonstationary is more complicated, because we needto add equation of first order (7.8.11) to nonlinear wave equation (7.8.17). Hence, theproblem is not “strongly” elliptic, and one needs to develop further the method oflower-upper solutions.

Lemma 7.12. Let �⊂ Rn—bounded domain with boundary ∂� ∈ ∈ C2,α , α ∈]0,1[.Let u0 ∈ C2,α(�) and h ∈ C0,1

loc (�×R) such that h(x, ·)—monotonic increasing func-tion for every x ∈�. Then boundary value problem

4u= h(·,u(·)) in �,

u= u0 on ∂�(7.8.39)

possesses a unique solution u ∈ C2,α(�).

Proof. Due to monotonicity of h, it is easy to check that there exist p1,p2 ∈ C0,α(�)

such that p2(x)≤ 0≤ p1(x) and

h(x,s)

{≤ p1(x) for s≤ 0,≥ p2(x) for s≥ 0

for all x ∈ �. Let u01 =min(u0,0) and u02 =max(u0,0). Let uk ∈ C2,α(�)—solutionof linear boundary value problem for k ∈ (1,2){

4uk = pk in �,

uk = u0k on ∂�.

Due to the maximum principle, u1 ≤ 0≤ u2 in �. From the last one follows that u1—a lower solution and u2—an upper solution for (7.8.39). Then from the theorem ofexistence (see Pao [224], Theorem 7.1]) follows, (7.8.39) has a unique solution u ∈C2,α(�).

Remark 7.7. Lemma (7.12) is a well-known statement and does not require additionalcomments. We remark only, on the condition of monotonicity of function h(x, ·) forthe VP system as applied first by Vedenyapin [284], [286].

Introduce the following conditions on function f : R→ [0,∞) :

(f1) f ∈ C1(R);(f2) ∀u ∈ R : f ∈ L1(u,∞);(f3) f is measurable function and f (s)≤ Ce−s for a.e. s ∈ R;(f4) f is decreasing, f (0)= 0 and ∃µ≥ 0 : ∀s≤ 0 : f (s)≤ C|s|µ.

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Lemma 7.13 (Braasch [55]). Let function f : R→ [0,∞) is given and

hf (u)= c∫R3

f (v2+ vd+ u), u ∈ R.

Then the following claims hold:

1. Assume conditions ( f2),( f3). Then hf : R→ R is continuous and nonnegative,

hf (u)=c1

|d|

∞∫1

|d|s2∫−|d|s2

sf (s+ t+ u)dtds

for all u ∈ R.2. Assume condition ( f3). Let ψ :→[0,∞)—measurable function and ψ ≤ f (a.e.). Then

hψ ≤ hf .3. Assume condition ( f4) and |d|< 1. Then the following conditions ( f2),( f3), hf —continuously

differentiable and hf (u)≤ Ce−u for all u ∈ R are satisfied.4. Assume ( f4) and |d|< 1. Then from ( f4) it follows that hf —is decreasing function and

|hf (u)| ≤ C|u|µ

for all u ∈ R, where C = C(µ, |d|).

Lemma 7.14. Let � ∈ R2 with a smooth boundary ∂� ∈ C1. Let f1, . . . , fn : R→[0,∞) satisfy conditions ( f1)—(f3) and |d|< 1. Let hf : �×R→ R is given by

hf (x,U)=−2πq

αm(4α2c2

− d2)

n∑i=1

qi

∫R3

fi(−αv2+ vdi+ liU(x, t))dv,

and we assume U ∈ C2(�)—solution of boundary problem{LU4=

∂2U∂t2− c24U = hf (·,U) in �,

U = 0 on ∂�.(7.8.40)

We define

U(x, t)= U(x+ td), t ≥ 0, x ∈ �,

K(x, t)4=−

d2

4α2c2− d2

(∇U(x, t)− |d|−2∂tU(x, t)d

), t ≥ 0, x ∈ �,

K ∈ C1([0,∞[×�)3 and E,B by means of (7.8.35), (7.8.36).Then ( f1, . . . , fn,E,B, ) is classical solution of the VM system in � and, it satisfies

boundary conditions (7.8.37), (7.8.38).

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Proof. Due to Lemma 7.13, hf —continuous and continuously differentiable function.The function U satisfies equation (7.8.11). Therefore, it follows from Theorem 7.8that f1 . . . , fn is a solution of the Vlasov equation, and E,B—solution of the Maxwellsystem. Since U vanish on ∂�, then from definition U and translation invariance � ind we obtain that U and ∂tU vanish on [0,∞)× ∂�. Hence, ∇U× n� = K× n� = 0on [0,∞)× ∂�. From the last one, one obtains

E(x, t)× n�(x)= (K(x, t)−∇U(x, t))× n�(x)= 0

and

B(x, t)× n�(x)= |d|−2(n�(x)×K(x, t))d = 0

at t ≥ 0 and x ∈ ∂�. Therefore, the boundary conditions (7.8.37) are satisfied.

Theorem 7.9. Let �⊂ R3. Let f1, . . . , fn : R→ [0,∞) satisfy condition (f1) andare (pointwisely) less than corresponding functions ψ1, . . . ,ψn : R→ [0,∞) satis-fying condition (f4) with µ > 0. We suppose that |d|< 1 and there exists functionU ∈ C2

C(�) such that

U(x, t)= U(x+ td), t ≥ 0, x ∈�.

Then (7.8.35) in Lemma 7.14 possesses a smooth solution and f1, . . . , fn generates theclassical solution ( f1, . . . , fn,E,B) of the VM system in� of the form (7.8.34)–(7.8.36).

Proof. Since elliptic operator L in (7.8.40) has constant coefficients, then by linearchange of coordinates, it is possible to transform to Laplace operator L=4. Introduce

notations F4= ( f1, . . . , fn), and we write the right part hF of (7.8.40) as

hF(x,U)=−c1(c2− d2)

n∑i=1

qihfi(liU(x)),

where functions hf1 , . . . ,hfn are defined in Lemma 7.14. From Lemmas 7.12 and 7.13we obtain

hF(x,U)

≥−c1(c2− d2)∑

qi>0 Ci|qi|hψi(|li|U(x))4= h1(x,U),

≤ c1(c2− d2)∑

qi<0 Ci|qi|(−|li|U(x))4= h2(x,U),

where hψ1 . . . ,hψn : R→ R—continuously differentiable, decreasing, nonnegativefunctions. Moreover, functions h1,h2—continuously differentiable and increasing inU and h1 ≤ 0≤ h2.

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7.9 Linear Stability of the Stationary Solutions ofthe Vlasov-Maxwell System

Introduce a set of the vector functions W = ( fi,E,B) with fi = fi(S), i= 1, . . . ,N, S=−αiV2

+ dV +F(r, t); E =∇F(r, t), B= c2E× d and denote it by means of S.Consider the question of the stability of a stationary solution W0 = ( f0i,E0,B0)

from a class S corresponding the fixed distribution functions fi(S). Let

||W0|| =

N∑

i=1

∫�1

∫�2

f 20idrdV

1/2

+

∫�2

B20dr

1/2

+

∫�2

E20dr

1/2

.

We define the solution

W(r,V, t)= {fi(r,V, t),E(r, t),B(r, t)}

of the VM system (7.1.2)–(7.1.6) with initial conditions corresponding to the samedistribution functions as the stationary solution W0(r,V). Then

fi|t=t0 = f 0i (−αiV

2+Vd+ F(r)), i= 1, . . . ,N,

E|t=t0 =∇F(r),

B|t=t0 = c2∇F(r)× d,

i.e.,

W(r,V, t)|t=t0 = W0(r,V), W0(r,V) ∈ S.

Let

E× n|∂�2 = 0, (7.9.1)

(B,n)|∂�2 = 0, (7.9.2)

then we have the following definition.

Definition 7.1. The stationary solution W0(r,V) from a class S is called Lyapunovstable if ∀ε > 0 and ∀W0 ∈ S, ∃δ = δ(ε,T) such that when the norm ||W0−W0||< δ,then the norm ||W(r,V, t)−W0||< ε for 0< t < T , where 0< T <∞.

The equilibrium configuration, which we tested on stability, represents the chargedelectron-ion bundle with nonrelativistic movement of particles confined in a cylinderwith a finite radius and retained by the magnetic field (see Davidson [83]). Moreover,

(d,n)= 0. (7.9.3)

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In cylindrical geometry (ρ,2,Z) the boundary conditions (7.9.1), (7.9.2), along with(7.8.24) and (7.9.3), are concretized

∂U

∂2|∂�2 = 0,

∂U

∂Z|∂�2 = 0.

Let Gi( fi) be smooth functions and

∫�1

Gi( fi)dV <∞,∫�1

fidV <∞,

then (7.1.2)–(7.1.6), describing the behavior of the electron-ion bundle in the cylinder,have the following first integrals by (7.9.1) and (7.9.2):

T =1

8π

∫�2

{E2+B2}dr+

N∑i=1

∫�1

∫�2

1

2miV

2fidrdV, (7.9.4)

F1 =

N∑i=1

∫�1

∫�2

Gi( fi)drdV,

F2 = (d,P), (7.9.5)

P=N∑

i=1

∫�1

∫�2

Pi fidrdV +1

4πc

∫�2

E×Bdr, Pi = miV.

Definition 7.2. (Holm, Marsden, Ratiu, Weinstein [137]) The stationary solution

W04= ( f0i,E0,B0) for the system of equations (7.1.2)–(7.1.6) is called formally stable

if there exists the Lyapunov functional L, which possesses an isolated minimum in astationary point W0.

If the second variation of functional L is strongly positive, then W04= ( f0i,E0,B0)

is an isolated minimum.Following Chetaev’s method, we introduce the Lyapunov functional in the form of

a bundle of the first integrals

L−L0 = T +F1+ λF2, (7.9.6)

where L0 is a functional value that is calculated along the nonperturbed (stationaryW0) state of the system and λ is an auxialiary parameter (Lagrange’s coefficient).

Calculate the first variation of functional (7.9.6) on variables fi,E,B. In addition,we restrict our consideration to the two-component (N = 2) system of (7.1.2)–(7.1.6).

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The first variation of energy integral (7.9.4), in the point of equilibrium, has thetype

δT =1

4π

∫�2

{E0δE+B0δB}dr+2∑

i=1

∫�1

∫�2

mi

2V2δfidrdV. (7.9.7)

Reduce the first subintegral expression in a functional (7.9.7) using the connection offields E,B with potentials ϕ and A

E =−∇ϕ−1

c

∂A

∂t, E0 =−∇ϕ0, (7.9.8)

B= rotA, B0 = rotA0 (7.9.9)

and gauge condition (7.8.30). After transformations, we have

δT =1

4π[∫�2

{ϕ0

(1

c2

∂2

∂t2δϕ−4δϕ

)−4A0δA

}dr−

∮∂�2

ϕ0(δE,n)dS +

+

∮∂�2

(δA×B0)ndS ]+2∑

i=1

∫�1

∫�2

1

2miV

2δfidrdV.

Similar calculations of the first variation for the momentum integral (7.9.5), takinginto account (7.9.8), (7.9.9), and (7.8.30), give

(d,δP)=−d

4πc

∫�2

{∇ϕ0∇δA+ϕ04δA+4A0δϕ+

1

c

∂

∂t[d× δA]rotA0

}dr+

+

∮∂�2

{ϕ0[n× δB]−ϕ0∇δA+ [n×B0]δϕ}dS

+ d2∑

i=1

∫�1

∫�2

PiδfidrdV

and the expression∫�2

{(d,∇ϕ0)∇δA+

1

c

∂

∂t[d× δA]rotA0

}dr

vanishes through the use of (7.8.26), (7.8.29).In addition, introduce the first integrals of the type

F3 =

∫�2

8(ϕ)dr, (7.9.10)

F4 =

2∑i=1

∫�1

∫�2

9i(x,y,z, t)fidrdV, (7.9.11)

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with twice differentiable function 8(ϕ) and function 9i satisfying the equation

∂9i

∂t+

1

2α(∇9i,d)= 0, (7.9.12)

where, by the terminology of Moisseev, Sagdeev, Tur, Yanovskii [208], the functions9i are Lagrangian invariants.

We show that the functionals (7.9.10) and (7.9.11) are really first integrals of(7.1.2)–(7.1.6) in view of the symmetry problem along d. Differentiate (7.9.10),(7.9.11) with time. Since the function 8(ϕ) satisfies the equation

∂8

∂t+

1

2α(d,∇8)= 0,

then, due to (7.8.11) and (7.8.29), we obtain

d

dt

∫�2

8dr =∫�2

∂8

∂tdr =−

1

2α

∫�2

(d,∇8)dr =−1

2α

∮∂�2

8(n,d)dS= 0.

Further

d

dt

2∑i=1

∫�1

∫�2

9i fidrdV =

=

2∑i=1

∫�1

∫�2

(∂

∂t9i fi−9i∇r(Vfi)−

qi

mi∇V ·9i

(E+

1

c[V ×B]

))drdV

=

=

2∑i=1

∫�1

∫�2

(∂9i

∂t+

1

2α(∇9i,d )

)fidrdV −

∮∂�2

∫�1

9i V fidV

dS −

−qi

mi

∮∂�1

∫�2

9i

(E+

1

c[V ×B]

)fidr

dS1

= 0.

Remark 7.8. The integral (7.9.10) admits a generalization of the type

F3 =

∫�2

8(ϕ,91, . . . ,9n)dr,

where the functions 9n are Lagrangian invariants of (7.9.12).

Consider a final structure of the Lyapunov functional by (7.9.10) and (7.9.11)

L= L−L0 = T +F1+ λF2+F3+F4. (7.9.13)

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After preliminary calculations, the first variation of functional (7.9.13) has the form

δL(δfi,δE,δB)=2∑

i=1

∫�1

∫�2

{mi

2V2+G′i( f0i)+ dλPi+90i+ qiϕ0

}δfidrdV−

−

∫�2

{1

4π4A0δA+ λ

d

4πc(ϕ04δA+4A0δϕ)−8

′(ϕ0)δϕ

}dr−

−1

4π

∮∂�2

{ϕ0(δE,n)− [δA×B]n}dS−

− λd

4πc

∮∂�2

{ϕ0[n× δB]−ϕ0∇δA+ [n×B0]δϕ}dS,

where λ=−1/2α. Assuming

ϕ0|∂�2 = 0 (7.9.14)

and taking into account (7.8.24), (7.8.29) we have

δL=2∑

i=1

∫�1

∫�2

{1

2miV

2+G′i( f0i)−

d

2αPi+90i+ qiϕ0

}δfidrdV +

+

∫�2

{8′(ϕ0)+

d2

16πα2c24ϕ0

}δϕdr (7.9.15)

with the condition

δϕ0|∂�2 = 0.

From equality to zero of the first variation (7.9.15) of functional (7.9.13), we have theequations of the equilibrium, which states

G′i( f0i)+1

2miV

2−

d

2αPi+90i+ qiϕ0 = 0, (7.9.16)

8′(ϕ0)+d2

16πα2c24ϕ0 = 0 (7.9.17)

with condition (7.9.14).The correlation (7.9.16) permits us to concretize a structure of the distribution func-

tions for which we can show stability. Introduce notation

G′i( f0i)=−H, H =

{1

2miV

2−

d

2αPi+90i+ qiϕ0

}. (7.9.18)

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A choice of the functions G′i( f0i) of type (7.9.18) implies

f0i(r,V)4= f0i(H).

Then by putting

G′i( f0i)=1

βiln

f0i

γβi, γ > 0 (7.9.19)

we concretize the distribution functions

f0i = γ exp(−βiH). (7.9.20)

From (7.9.19) we have

Gi( f0i)=1

βi{ f0iln f0i− f0i− f0ilnγ }. (7.9.21)

Analysis of the equations of equilibrium states (7.9.16), (7.9.17) and stationary equa-tions (7.8.25), (7.8.26) define the values of parameter βi in (7.9.20) and functions 90i

βi =2αi

mi, 90i =−

qi

2αc(d,A0). (7.9.22)

Moreover, on the basis of (7.8.29) and (7.9.17), we obtain

d2mc

8παq(d2− 4α2c2)4U0+8

′(U0)= 0,

U0|∂�2 =−d2

4α2c2ϕ0|∂�2; 4ϕ

0= 0; ϕ0

|∂�2 = b(r), (7.9.23)

where b(r) is a given function. The harmonic function ϕ0 in (7.9.23) may be discussedas a given external field on the boundary ∂�2. Further, it is easy to write the sufficientconditions of positive definiteness of the initial Lyapunov functional (7.9.13) takinginto account (7.9.21) and (7.9.22)

|d|

2α< c; ln f0i ≥ 1+ ln|γ |, (7.9.24)

2∑i=1

{⟨1

2miV

2−

d

2αPi

⟩+9i(r)

}> 0, (7.9.25)

<a>4=

∫�1

a(V)f (r,V)dV∫�1

f (r,V)dV,

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or

2∑i=1

⟨1

2miV

2−

d

2αPi

⟩>

2∑i=1

qid2

4α2c2

(ϕ0−

m

2αqϕ0).

Since d/2α =< V >= V −VT , where VT is a mean of random heat velocity, then(7.9.25) becomes

2∑i=1

{⟨1

2miV

2T

⟩−

⟨1

2mid

2⟩}>

2∑i=1

qid2

4α2c2

(ϕ0−

m

2αqϕ0). (7.9.26)

The condition of (7.9.26) places a restriction on the value of fields in the system,moreover, d2/c2

� 1.The second variation of the functional (7.9.13) has the form

δ2L=1

4π

∫�2

{(δE)2+ (δB)2}dr+1

4απc

∫�2

(d,δB× δE)dr+

+

2∑i=1

∫�2

∫�1

G′′i ( f0i)(δfi)2dVdr+

∫�2

8′′(ϕ0)(δϕ)2dr. (7.9.27)

It is easy to show that, taking into account (7.9.1)–(7.9.3), the second variation of theLyapunov functional (7.9.27) is an integral of a linearized VM system.

Sufficient conditions for the stability of equilibrium solutions (7.9.16), (7.9.17) canbe obtained from the condition of the positive definiteness of the subintegral expres-sion in formula (7.9.27). For the positive definiteness of δ2L in the neighborhood of astationary state, it is sufficient that we have the following conditions:

|d|

2α< c; G′′i ( f0i) > 0; 8′′(ϕ0) > 0,

or by (7.9.16)

|d|

2α< c;

∂f0i(H)

∂H< 0; 8′′(ϕ0) > 0, i= 1,2.

Using the stability theorem [262], we obtain the sufficient conditions of stability forthe stationary (equilibrium) solutions (7.9.16) and (7.9.17) by measure ρ. As a mea-sure by which the stability is studied, we choose the quantity

ρ = ||δE||2L2(�2)+ ||δB||2L2(�2)

+

2∑i=1

||δfi||2L2(�1×�2)

+ ||δϕ||2L2(�2). (7.9.28)

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Let the potential ϕ0 and a solution ϕ0 of the linear boundary—value problem (7.9.23)satisfy the inequalities (7.9.24) and (7.9.26), then, for the stability of stationary (equi-librium) solutions (7.9.16) and (7.9.17) by measure (7.9.28), it is sufficient that thefollowing conditions are satisfied

|d|

2α< c, 0< Ci ≤ G′′i ( f0i)≤ bi, bi > 0, Ci < bi,

0< l1 ≤8′′(ϕ0)≤ l2, l2 > 0, l1 < l2; Ci,bi, l1, l2−Const,

or the conditions

|d|

2α< c,

−1

Ci≤∂f0i(H)

∂H≤−

1

bi< 0,

0< l1 ≤8′′(ϕ0)≤ l2.

7.10 Application Examples with Exact Solutions

Taking into account distribution (7.6.1), the solution of the equation (7.5.31) reducesto the expression

4u(ξ,η)= a(d,α)N∑

i=1

qi

(π

αi

)3/2

exp

(γi+

d2i

4αi

)eliu. (7.10.1)

We assume N = 2;q1 < 0,q2 > 0, i.e., f1(r,v), f2(r,v) distribution functions of ionsand electrons respectively, defined by formulas (7.6.1). Taking arbitrary constant γ2 inthe form

γ2 = γ1+1

4

(d2

1

α1−

d22

α2

)+ln

[|q1|

q2

(α2

α1

)3/2],

we will obtain an equation of the sh-Gordon type

4u= ω(eu− elu), l ∈ R−, (7.10.2)

where

w(d1,α1,γ1)= |q1|

(π

α1

)3/2

a(d,α)exp

(γ1+

d21

4α1

).

Let us consider the construction of some exact solutions of equation (7.10.2). Itfollows from (7.6.1) that a value αi/mi is proportional to temperature of i compo-nent of plasma. Thus, for the concrete definition, we assume that temperatures of the

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components of plasma are equal α1/m1 = α2/m2. We use coonection of the charges q1and q2 : q2 =−Zq1, where Z = 1, . . . ,N. If Z = 1 (the case of the completely ionizedhydrogen plasma), then l=−1, and (7.10.2) takes the form

4u(ξ,η)= 2wsinhu(ξ,η). (7.10.3)

Assuming (see [218])

u(ξ,η)= 2 ln

∣∣∣∣(X(ξ)+Y(η))/(X(ξ)−Y(η))

∣∣∣∣, (7.10.4)

we transform (7.10.3) into the system of ODEs

(X′)2 = m2X4− (n2

−w)X2+ k2,

(Y ′)2 =−m2Y4+ n2Y2

− k2.(7.10.5)

Here m2, n2, and k2 are arbitrary parameters. For m2= 0, n2

6= 0, k26= 0, n2

−w>0there is a partial solution

X(ξ)= sin[ξ(n2−w)1/2]k(n2

−w)−1/2,

Y(η)= cosh(nη)kn−1.(7.10.6)

As an example, Figures 7.1–7.5 are several 3D graphs of solution (7.10.6).From these graphs it is possible to note that the solution possesses the periodic

structure, connected with the phenomenon of magnetic islands for the infinite plasma.

Figure 7.1 Partial solution for (7.10.6) for n= 2, ω = 2.77, k = 34 , ξ =−π . . .π, µ=−0.6

. . .0.6.

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Figure 7.2 Partial solution for (7.10.6) for n= 2, ω = 2.77, k = 34 , ξ =−π . . .π, µ=−0.6

. . .0.6.

Figure 7.3 Partial solution for (7.10.6) for n= 2, ω = 2.77, k = 34 , ξ = 0.695π . . .0.71π, µ=

0.11 . . .0.175.

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Figure 7.4 Partial solution for (7.10.6) for n= 2, ω = 2.77, k = 34 , ξ = 0.69727π . . .

0.69862π, µ= 0.173 . . .0.174.

Substituting (7.10.6) into (7.10.4), inverting variables by the formulas (7.5.21) andsolving linear Dirichlet problem (7.5.32), it is easy to reconstruct fields E,B in domain� and distribution functions f1, f2 by Theorem 7.2.

We construct some exact solutions of sh-Gordon equation (7.10.3), using Xirotamethod [134]. According to this method, we will find solution in the form

u= 2ln

∣∣∣∣F+G

F−G

∣∣∣∣, (7.10.7)

where F and G are the functions of ξ and η. Substituting (7.10.7) into (7.10.3), weobtain equations for definition F and G

(F2+G2)D2

4F ◦G−FGD2

4(F ◦F+G ◦G)= FG(F2

+G2)2w. (7.10.8)

Here let D24

be bilinear Hirota operator, acting in the following way:

D24

F ◦G= (D2ξ +D2

η)F ◦G4=

4=

[(∂

∂ξ−

∂

∂ξ ′

)2

+

(∂

∂η−

∂

∂η′

)2]F(ξ,η)G(ξ ′,η′)

∣∣∣∣∣ξ=ξ ′, η=η′

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Figure 7.5 Partial solution for (7.10.6) for n= 2,ω = 2.77,k = 34 ,ξ = 0.697711π . . .

0.697712π,µ= 0.17329892 . . .0.173299.

or

D24

F ◦G= F4G+G4F− 2∇F∇G

and hence,

D24

F ◦F = 2(F4F− (∇F)2).

To solve (7.10.8) let

D24

F ◦G= FG(2w), D24(F ◦F+G ◦G)= 0.

We reduce the last system to one equation by change of variables F = ( f + f )/2,G= ( f − f )/2

D24

f ◦ f =1

2( f 2− f 2)(2w).

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One can solve this equation by means of the choice of functions f and f in the form

fN =√

2∑µ=0,1

exp

N∑i=1

µiηi+∑

1≤i≤j

µiµjAij

sin

N∑i=1

µiπ/2+π/4

,fN =−

√2∑µ=0,1

exp

N∑i=1

µiηj+∑

1≤i≤j

µiµjAij

sin

N∑i=1

µiπ/2−π/4

,(7.10.9)

N = 1,2,3. Here sum in µ passes on all sets, Aij = lnaij, aij4= (ki− kj)

2/(ki+ kj)2,

ki = (kiξ ,kiη) arbitrary vectors, normed by conditions k2i = 2w, ηi

4= ki(r− r0), r0—

constant vector, i, j= 1, . . . ,N. Proof of (7.10.9) is carried out by the standard reason-ings.

As example, let us consider solutions of Sinh-Gordon equation for N = 1,2. ForN = 1 we obtain

u(ξ,η)= lncotanh2[k(r− r0)/2].

For N = 2 from (7.10.9) follows that

u= 2 lna1/2

12 sinh{

12 [(k1+ k2)(r− r0)+A12]

}− cosh

{12 (k1− k2)(r− r0)

}a1/2

12 sinh{

12 [(k1+ k2)(r− r0)+A12]

}+ cosh

{12 (k1− k2)(r− r0)

} .(7.10.10)

Solution (7.10.10) is illustrated by graphs 7.6–7.10 with the following values of thevectors k1 = [0,2], k2 = [2,0].

Vectors k1 and k2 can be complex values. If, moreover, k1 = k2, then u(ξ,η) is real.If we assume k1 = a+ ib, then a2

− b2= 2w, a · b= 0, then

u(ξ,η)= 2lna1/2

12 sinh[a(r− r0)+A12/2]− e(1/2)a(r−r0)cos b(r− r0)

a1/212 sinh[a(r− r0]+A12/2]+ e(1/2)a(r−r0)cos b(r− r0)

. (7.10.11)

Figures 7.11–7.14 demonstrate graphical solution of (7.10.11). Here orthogonalvectors a,b are a= [−1.2;1.2],b= [0.3;−4.2].

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Figure 7.6 Partial solution for (7.10.10) for x=−2 . . .2,y=−2 . . .2.

Figure 7.7 Partial solution for (7.10.10) for x=−2 . . .0,y=−2 . . .0.

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Figure 7.8 Partial solution for (7.10.10) for x=−1.75 . . .− 0.15,y=−1.25 . . .− 0.15.

When constructing other real solutions of (7.10.3), it is necessary that the vectorski would satisfy some additional relationships. Let us note that solution (7.10.3) givenabove is also valid in the 3D- case.

Consider equation (7.10.2) in case Z = 2 (ionized helium); l=−2 and (7.10.2)becomes

4u= w(d,α,γ )(eu− e−2u)

with the corresponding solution

u(ξ,η)= ln

[1−

3

2cosh2( 12 w1/2kr)

]−2

,

which is illustrated by Figures 7.15a, 7.15b, and 7.16.Further, we consider application of Theorem 7.3 to the system (7.6.2). In this case

condition (A) will be satisfied, if

2|q1|α−3/2ed2/4α+γ

= τ

N∑i=2

qiα−1/2i ed2

i /4αi+γi . (7.10.12)

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Figure 7.11 Partial solution for (7.10.11) in complex case for x=−2 . . .20,y=−2 . . .20.

Taking into account (7.10.12), the “resolving” equation (7.5.33) takes the form

4u= meu, m= εb

a+ εb

(π

α

)3/2

ed2/4α+γ . (7.10.13)

As an example of a partial solution, we can take

u= ln{α/4

[1/sinh2

(√α/2x+ c

)+ 1/cos2

(√α/2y+ c

)]}shown on Figure 7.17.

Remark 7.9. If it is necessary to find the solution of system (7.5.1)–(7.5.5) withadditional condition of normalization (6.1.6), then we take

eγi =

(αi

π

)3/2

e−d2i /4αi

∫�

eliudx

−1

,

in formula (7.10.1) for Case 1 Theorem (7.2), and we obtain integro-differential equa-tion. Equation (7.10.1) has a constant solution u= lnC, where C is defined fromalgebraic equation

N∑i=1

qi

(π

αi

)3/2

exp

(γi+

d2i

4αi

)Cli = 0.

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Figure 7.12 Partial solution of for (7.10.11) in complex case for x=−2.5 . . .2.5,y=−2.5 . . .2.5.

Figure 7.13 Partial solution for (7.10.11) in complex case for x= 10 . . .20,y= 10 . . .20.

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Figure 7.14 Partial solution of for (7.10.11) in complex case for x= 10 . . .20,y= 10 . . .20.

(a) (b)

Figure 7.15 Partial solution for k1 = k2 = 0.05; a) ω = 1.0,x=−20 . . .10,y=−20 . . .10;b) ω = 0.1,x=−12 . . .10,y=−12 . . .10.

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Figure 7.16 Partial solution for k1 = k2 = 0.005,ω = 0.5,x=−75 . . .75,y=−75 . . .75.

Figure 7.17 Partial solution for α = 4,m= 1,c= 1,x=−10 . . .10,y=−3 . . .4.

If parameters γi are assigned so that

N∑i=1

qiα−3/2exp

[γi+

d2i

4αi

]= 0,

then we can take C = 1 that corresponds a trivial solution u= 0.

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For case 2 Theorem (7.3) under condition (7.10.12) and coefficient m inequation (7.10.13),

eγi =

(αi

π

)3/2

e−d2i /4αi

∫�

eudξdη

−1

, i= 1, . . . ,N.

Therefore, we have condition

2q1+ τ

N∑i=1

qi = 0,

and the related Liouville equation becomes

4u= εb

a+ εb

eu∫�

eudξdη.

Equations of such type have been studied by Dancer [82] using methods of branch-ing theory.

We note that the general approach to find exact solutions of equations consideredin this section is given in monography [7].

7.11 Normalized Solutions for a One-ComponentDistribution Function

Consider the distribution function of type (7.6.1) for N = 1. In this case, the system(7.6.2) takes the form

4ϕ(r)=8παq2

mp(d,α)eϕ+ψ , (7.11.1)

−4ψ(r)=2πq2d2

αmc2p(d,α)eϕ+ψ , (7.11.2)

where

p(d,α)=

(π

α

)3/2

ed2/4α

with the conditions

(∇ϕ,d)= 0, (∇ψ,d)= 0. (7.11.3)

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If we substract the respective sides of (7.11.1) from both sides of (7.11.2), we thenobtain the Liouville equation

48(r)= a(d,α)d2− 4α2c2

αc2e−8, (7.11.4)

with

8=−ϕ−ψ; a(d,α)=2πq2

mp(d,α).

By determining the function 8 from (7.11.4) and substituting it into (7.6.1) forN = 1, we obtain the desired distribution function f . By knowing 8, we are able todetermine ∇ϕ and ∇ψ from (7.11.1) and (7.11.2) and, hence, to construct the desiredfields E and B by means of (7.5.7) and (7.5.11) by N = 1. By virtue of (7.11.3), it isnecessary to find only the solutions of (7.11.4) of the form

8(r)4=8

(x

d1−

y

d2,

y

d2−

z

d3

). (7.11.5)

Let us consider the solution of (7.11.4) of the form 8(S) with

S=x

d1−

y

d2+ k

(y

d2−

z

d3

), k = Const.

This class of solutions presents a most comprehensive investigation, because it is easyto construct E,B, f in an explicit form. Indeed, the corresponding solution 8(S) of(7.11.4) satisfied the ordinary differential equation

8′′(S)= b(d,α)e−8, (7.11.6)

where

b(d,α)= a(d,α)d2− 4α2c2

αc2

(1

d21

+(k− 1)2

d22

+k2

d23

)−1

.

Consider two cases:

1. d2− 4α2c2 > 0 and

2. d2− 4α2c2 < 0.

Note, the case when d2− 4α2c2

= 0 is trivial, because b(d,α)= 0, 8(S)= S0S+ S1,where S0,S1—Const. Therefore, f = exp(−αV2

+Vd− S0S− S1) and the constantsS0,S1,K and vector d are chosen from the normalization condition (6.1.6) by N = 1(

π

α

)3/2

exp(d2/4α)∫�2

exp(−S0S− S1)dxdydz= 1.

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Let d2− 4α2c2 > 0. In this case, the general solution of (7.11.6) is determined from

the relation

e−8 =2S2

1

b(d,α)cosh2S1(S− S0). (7.11.7)

Substituting (7.11.7) into (7.6.1) for N = 1, we have

f =2S2

1

b(d,α)cosh2S1(S− S0)exp(−αV2

+Vd)U. (7.11.8)

On the basis of (6.1.6), for the distribution function (7.11.8), the normalization condi-tion has the form

2S21

b(d,α)

(π

α

)3/2

exp(d2/4α)∫�2

dxdydz

cosh2S1(S− S0)= 1. (7.11.9)

Since �2 = R3, then at k = 1,k = 0 and the integral in (7.11.9) diverges. If k 6= 1,k 6=0, then this integral is calculated, and the relationship (7.11.9) can be transformed

|S1| =|d1d2d3|πmαc2

12|k(k− 1)|q2(d2− 4α2c2)K, K =

(1

d21

+(k− 1)2

d22

+k2

d23

).

(7.11.10)

The relationship (7.11.10) is a condition relating the parameters α,d and the integra-tion constant S1. Note, when �1 = R3,�2 = R3 the normalization condition (6.1.6)for the distribution function (7.11.8) can also be ensured on account of the parameter

α = (−l+ (l2+ 16h2c2d2)1/2)/8hc2 > 0

with

l= |d1d2d3|πmc2K; h= 12|k(k− 1)||S1|.

For determining the corresponding fields E,B from (7.5.7) and (7.5.11), it is enoughto know ∇ϕ and ∇ψ . In our case

∇ϕ = ϕ′(S)K, ∇ψ = ψ ′(S)K, K=

∣∣∣∣ 1

d1,

k− 1

d2,−

k

d3

∣∣∣∣′ (7.11.11)

and, due to (7.11.1), a function ϕ(S) satisfies the equation

ϕ′(S)=8α2c2S1

(d2− 4α2c2)tanhS1(S− S0)+ S2, (7.11.12)

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where S2- const and a function ψ(S) by (7.11.7)

ψ ′(S)=−2S1d2

(d2− 4α2c2)tanhS1(S− S0)− S2. (7.11.13)

Using (7.11.11)–(7.11.13) from (7.5.7), (7.5.11), we determine the fields

E =m

2αq

(8α2c2S1

(d2− 4α2c2)tanhS1(S− S0)+ S2

)K,

B=γ

d2d+

mc

qd2

[d×

(2S1d2

(d2− 4α2c2)tanhS1(S− S0)+ S2

)]K.

Let d2− 4α2c2 < 0, then (7.11.6) takes the form

F′′(S)= |b(d,α)|eF, (7.11.14)

where F =−8. From (7.11.14), it follows that∫dF

(2|b(d,α)|eF + 2c1)1/2+ c0 = S.

In this case, it is easy to see that the general solution of (7.11.14) is determineddepending on the sign of the constant c1, by either

eF=−

c1

2|b(d,α)|cos2√|c1|/2(S− c0)

, c1 < 0, (7.11.15)

or

eF=

c1

|b(d,α)|sinh2√c1/2(S− c0), c1 > 0. (7.11.16)

By determining from (7.11.15) the function F and taking into account (7.6.1), it is easyto verify that

∫R3×R3 f drdV =∞. From (7.6.1), subject to (7.11.16), we have

f = expF · exp

{−α

[(Vx−

d1

2α

)2

+

(Vy−

d2

2α

)2

+

(Vz−

d3

2α

)2]}. (7.11.17)

From the norming condition (6.1.6) for (7.11.17) when �1 = R3, �2 = R3, it followsthat the parameters α > 0, k 6= 0, k 6= 1, d ∈ R3 and the integration constants c1 > 0,c0 must satisfy the relation

2π |d1d2d3|p(d,α)

3√

2c1|k(k− 1)||b(d,α)|= 1.

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In this case, the electric E and magnetic B fields are determined in a similar fashion asabove.

When seeking the solution of the Liouville equation (7.11.4) in the form of (7.11.5),we have supposed that d1 6= 0,d2 6= 0 and d3 6= 0. In this case, the function 8(x,u),u= y/d2− z/d3 satisfies the elliptic equation

8xx+

(1

d22

+1

d23

)8uu = a(d,α)

d2− 4α2c2

αc2e−8.

By solving this, it is possible to use previous results. If d1 = d2 = 0 and d3 6= 0, thenaccording to (7.11.5), the function 8(x,y) satisfies the Liouville equation

48(x,y)= a(d,α)d2− 4α2c2

αc2e−8.

Note, in the case when�1 = R3 and�2 = R3, the distribution function is not normal-ized, because the integral in (6.1.6) will diverge.

Let d1 = d2 = d3 = 0 (this case corresponds to a cold plasma). Then from (7.5.8),(7.5.9) and (7.6.1) for N = 1, it accordingly follows

ψ(r)= 0− const, E =m

2αq∇ϕ, f = exp{−αV2

+D(r)} (7.11.18)

and from (7.11.1) we obtain

4D= ε(α)eD (7.11.19)

with

D=42+ϕ, ε(α)=8παq2

m

(π

α

)3/2

.

On the other hand, from (7.5.3), (7.5.5) N = 1, we have

divB= 0, rotB=4πq

ceD∫�1

Ve−αV2dV. (7.11.20)

If �1 ⊆ R3 is a finite or infinite symmetric domain, then from (7.11.20) it follows thatB=−∇3,43= 0, i.e.,3 is a scalar harmonic function. If D and3 are sought in theform D(S),3(S) with S= a1x+ a2y+ a3z, then by similar reasoning as above, from(7.11.18)–(7.11.20) we have

f =2S2

1

(a2

1+ a22+ a2

3

)ε(α)sinh2S1(S− c0)

exp(−αV2),

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E =mS1

αqcothS1(S− c0)

∣∣∣∣∣∣a1a2a3

∣∣∣∣∣∣ , B=−h0

∣∣∣∣∣∣a1a2a3

∣∣∣∣∣∣ ,and, in this case,3= h0S+ h1, where c0, S1, h0, h1, a1, a2, a3 are arbitrary constants.If�⊆ R3 is an arbitrary domain, then, through a choice of the parameters involved inthe distribution function f , it is possible to fulfill (6.1.6).

Consider the case when the domain �1 ⊂ R3 is not a symmetric one. From(7.11.20) subject to (7.11.19), we have

divB= 0, rotB= U(r), (7.11.21)

where

U =mS2

1(a21+ a2

2+ a23)

αcq(π/α)3/2sinh2S1(S− c0)

∫�1

Vexp(−αV2)dV

is the given vector. Relations (7.11.21) lead to the equalities

B= rotA, 4A=−U, (7.11.22)

if the vector potential A satisfies the Lorentz condition divA= 0. Subsequently, weshall suppose �1 = R+×R+×R+. If the components of the vector potential Aare sought in the form Ax(S),Ay(S),Az(S), where S= a1x+ a2y+ a3z, then using(7.11.22) we obtain

Ax = Ay = Az = µ(α)ln|sinhS1(S− c0)| + u0S+ u1,

B= (S1µ(α)cothS1(S− c0)+ u0)

∣∣∣∣∣∣a2− a3a3− a1a1− a2

∣∣∣∣∣∣ (7.11.23)

under the condition that the coefficients a1,a2,a3 satisfy the relation a1+ a2+ a3 = 0,where µ(α)= (π)−1/2(α)−3/2m/(8cq); u0,u1—Const. Therefore, we have

E =mS1

αqcothS1(a1x+ a2y− (a1+ a2)z− c0)

∣∣∣∣∣∣a1a2

−(a1+ a2)

∣∣∣∣∣∣ ,

f =2S2

1(a21+ a2

2+ (a1+ a2)2)

ε(α)sinh2S1(a1x+ a2y− (a1+ a2)z− c0)e−αV2

. (7.11.24)

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Note, by the choice of the parameters involved in the f , it is easy to achieve forit the fulfillment of the norming condition (6.1.6). For example, if �2 = R3, �1 =

R+×R+×R+, S1 > 0, a1 > 0 and a2 > 0, then the distribution function has the form(7.11.24)

S1 =(a2

1+ a1a2+ a22)πm

96αq2a1a2|a1+ a2|.

Remark 7.10. Formula (7.11.23) defines all solutions of system (7.11.21) of the formB= B(S). If a1+ a2+ a3 6= 0, then the system of equations (7.11.21) does not have asolution of such a form. Indeed, by substituting B(S) into (7.11.21), we get the systemof ordinary differential equations

AB(S)= g(S), (7.11.25)

where

A=

∣∣∣∣∣∣∣∣0 −a3 a2a3 0 −a1−a2 a1 0a1 a2 a3

∣∣∣∣∣∣∣∣ , g(S)=

∣∣∣∣∣∣∣∣Ux

Uy

Uz

0

∣∣∣∣∣∣∣∣ .The system (7.11.25) has the solution if and only if the solvability condition

(g, l)= 0, where A∗l= 0 is satisfied. If a1 6= 0, then l= (1,a2/a1,a3/a1,0)′, if a2 6= 0,then l= (a1/a2,1,a3/a2,0)′ and finally if a3 6= 0, thenl= (a1/a3,a2/a3,1,0). Thus, let a1+ a2+ a3 = 0, then the general solution of(7.11.25) is determined by the formula

B(S)=∫

A∗g(S)dS+ c, c= (c1,c2,c3).

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8 Bifurcation of Stationary Solutionsof the Vlasov-Maxwell System

8.1 Introduction

The problem of the bifurcation analysis of a VM system, formulated for the firsttime by Vlasov, proved to be very complicated against the general background ofthe progress of the bifurcation theory in other directions, and it remains open at thepresent moment. There exist only separate results. One simple theorem about the pointof bifurcation is covered by Sidorov and Sinitsyn [257], and another is proven inpaper [256] for the stationary VM system.

The goal of this chapter is to prove the general existence theorems for the potentialsof electromagnetic field and for the density of charge and current; to find bifurcationpoints of stationary VM system with the given boundary conditions. For studyingthe bifurcation points of VM system are used the results of the branching theory ofTrenogin, Sidorov [272], Vajenberg, Trenogin [279] and the index theory of Conley[75], Kronecker [163].

Let us note that the methods, which were adapted in [256, 257], were not sufficientfor examination of the situation in general, which we study below.

Let us consider many component plasma consisting of electrons and positivelycharged ions of various kinds, described by many particle distribution functions ofthe form fi = fi(r,v), i= 1, . . . ,N. Plasma is located in domain D⊂ R3 with a smoothboundary. Particles interact among themselves only by means of their own charges;we neglect collisions between particles.

The behavior of plasma is described by the following (classical) version of station-ary VM system [305]:

v · ∂rfi+ qi/mi

(E+

1

cv×B

)·∂vfi = 0, (8.1.1)

r ∈ D⊂ R3, i= 1, . . . ,N,

curlE = 0,

divB= 0,


http://dx.doi.org/10.1016/B978-0-12-387779-6.00008-9

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divE = 4πN∑

k=1

qk

∫R3

fk(r,v)dv4= ρ, (8.1.2)

curlB=4π

c

N∑k=1

qk

∫R3

vfk(r,v)dv4= j.

Here ρ(r), j(r) are the densities of charge and current, and E(r), B(r) are electric andmagnetic fields, respectively.

We look for the solution E, B, f of the VM system (8.1.1), (8.1.2) for r ∈ D⊂ R3

with boundary conditions on potentials and densities

U |∂D = u01, (A,d) |∂D = u02; (8.1.3)

ρ |∂D = 0, j |∂D = 0, (8.1.4)

where E =−∂rU, B= curlA, and U, A scalar and vector potentials.We call a trivial solution E 0, B 0, f 0 if ρ 0

= 0 and j 0= 0 inside domain D.

Here we study the case of the distribution functions of the special form [248]

fi(r,v)= λfi(−αiv2+ϕi(r), v · di+ψi(r))

4= λfi(R,G), (8.1.5)

ϕi : R3→ R, ψi : R3

→ R, r ∈ D⊆ R3, v ∈ R3,

λ ∈ R+, αi ∈ R+ 4= [0,∞), di ∈ R3, i= 1, . . . ,N,

where functions ϕi, ψi, which generate appropriate electromagnetic field (E, B),should be found.

We are interested in the dependence of unknown functions ϕi, ψi on parameter λin distribution function (8.1.5). First we consider λ, which does not depend on phys-ical parameters αi, di used in (8.1.5). For example, in case αi = αi(λ), di = di(λ) thedistribution function

f (r,v)=

(m

2πkT

)3/2

·exp

(−m|v|2

2kT+ d · v+ϕ(r)

)gives a dependence α = α(λ), where λ= (m/(2πkT))3/2, α =−m/(2kT), k—Boltzmann constant, and T—temperature of electrons. In this case parameter, λ hasa dimension of the temperature.

Definition 8.1. A point λ0 is called the bifurcation point of VM system with condi-tions (8.1.3), (8.1.4), if in any neighborhood of vector (λ0,E 0,B 0, f 0) correspond totrivial solution with ρ0

= 0, j0 = 0 in domain D: there exists a vector (λ,E, B, f ),satisfying (8.1.1)–(8.1.4) for which

‖ E−E 0‖ + ‖ B−B 0

‖ + ‖ f − f 0‖> 0.

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Bifurcation of Stationary Solutions of the Vlasov-Maxwell System 135

Let ϕ0i , ψ0

i be constants with corresponding densities ρ0 and j0, introduced inmedium by distributions fi which are equal to zero in D for ϕ0

i , ψ0i . Then

N∑k=1

∫R3

qk fkdv= 0,N∑

k=1

∫R3

qkvfkdv= 0

for ϕi = ϕ0i and ψi = ψ

0i , k = 1, . . . ,N. Let di = σid1, i= 1, . . . ,n and let σi be a con-

stant. Vector di ∈ R3 and parameter αi ∈ R+ characterize the chaotic heat motion ofparticles of the kind i. We examine the case, when di and αi are different, i.e., non-isothermic plasma as most frequently being encountered in the applications. Then VMsystem possesses a trivial solution

fi0= λfi(−αiv

2+ϕ0

i , v · di+ψ0i ), E 0

= 0, B 0= βd1, β −Const.

for any λ.Our aim is to construct nontrivial solutions of the stationary VM system. We obtain

conditions implying the existence of points λ∗ ∈ R+ (bifurcation points), process-ing a neighborhood, where a VM system has nontrivial solutions in domain D⊂ R3.For these solutions we have ρ|D 6= 0, j|D 6= 0 but ρ|∂D = 0, j|∂D = 0. It is assumedthat the scalar and vector potentials of the desired electromagnetic field are given atthe boundary of the domain. The branching equation (BEq) was derived by Vain-berg and Trenogin in [274, 279]. We proved that for sufficiently general case offi = fi(a(−αiv2

+ϕi)+ b(v · di+ψi)), where a,b are constants, BEq be the potentialequation. On this basis the asymptotics are constructed for nontrivial branches of solu-tions in a neighborhood of the bifurcation point.

Let us note that the problem of bifurcation points in the theory of collision-less plasma without allowance for magnetic field was studied in Holloway [135],Holloway, Dorning [136], and Hesse and Schindler [131]. Apparently the problemof bifurcation points for the general VM system was not considered earlier.

The chapter is organized as follows: In Section 8.2, two existence theorems of thebifurcation points for the nonlinear operator equation in Banach space, generalizingresults on bifurcation points in [255, 272], are proven. The method of proof of thesetheorems uses index theory of vector fields [75, 163] and makes it possible to studynot only points, but also bifurcational surfaces with the minimum limitations to theequation.

In Section 8.4, we reduced the problem about a bifurcation point of a VM system tothe problem on bifurcation point of semilinear elliptic system, considered as operatorequation in Banach space. The boundary value problem and the problem on bifurca-tion points was formulated, and a spectrum of the problem for linearized system isstudied.

In Section 8.5, the BEq is constructed.In Section 8.6, the existence theorem for bifurcation points is proven on the basis

of the analysis of the BEq, and the asymptotics of nontrivial branches is constructedfor the solutions of the VM system.

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8.2 Bifurcation of Solutions of Nonlinear Equationsin Banach Spaces

Let E1, E2 be Banach spaces; ϒ be normed space. Let us consider equation

Bx= R(x,ε). (8.2.1)

Here B : D⊂ E1→ E2 be a closed linear operator with dense domain of definition inE1. Operator R(x,ε) with the values into E2 is defined, continuous and continuouslydifferentiable in Frechet sense over x in the neighborhood

�= {x ∈ E1, ε ∈ ϒ : ‖ x ‖< r, ‖ ε ‖< %}.

Assume R(0,ε)= 0, Rx(0,0)= 0. Let operator B be Fredholm operator. Let us intro-duce basis {ϕi}

N1 in subspace N(B), basis {ψi}

N1 in N(B∗), and also systems {γi}

N1 ∈ E∗1 ,

{zi}N1 ∈ E2, which are biorthogonal to these bases.

Definition 8.2. A point ε0 is called the bifurcation point of (8.2.1), if in any neighbor-hood of point x= 0, ε0 there is a couple (x,ε) with x 6= 0, satisfying equation (8.2.1).

It is known [279] that the problem about bifurcation point of equation (8.2.1) isequivalent to the problem on bifurcation point of finite-dimensional system

L(ξ,ε)= 0, (8.2.2)

where ξ ∈ RN , L : RN×ϒ→ RN . We will call (8.2.2) the branching equation (BEq).

We write 8.2.1 as the system

Bx= R(x,ε)+N∑

s=1

ξszs, (8.2.3)

ξs =< x,γs >, s= 1, . . . ,n, (8.2.4)

where Bdefn= B+

∑Ns=1 < ·,γs > zs has it inverse bounded. Equation (8.2.3) has a

unique small solution

x=N∑

s=1

ξsϕs+U(ξ,ε) (8.2.5)

for ξ → 0, ε→ 0. Substitution (8.2.5) into (8.2.4) gives formulas for coordinates ofvector-function L : RN

×ϒ→ RN

Lk(ξ,ε)=

⟨R

(N∑

s=1

ξsϕs+U(ξ,ε),ε

),ψk

⟩. (8.2.6)

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Here the derivatives

∂Lk

∂ξi|ξ=0 =

⟨Rx(0,ε)(I−0Rx(0,ε))

−1ϕi,ψk

⟩defn= aik(ε)

are continuous in neighborhood of point ε = 0, ‖ 0Rx(0,ε) ‖< 1.Introducing the set�= {ε | det[aik(ε)]= 0} containing the point ε = 0, we are able

to formulate the following condition:Condition (A) We assume that there exists the set S, in a neighborhood of

point ε0 ∈�, which posesses Jordan continuum S= S+⋃

S−, ε0 ∈ ∂S+⋂∂S−. More-

over, there is continuous mapping ε(t), t ∈ [−1,1] such that ε : [−1,0)→ S−, ε :(0,1]→ S+, ε(0)= ε0, det[aik(ε(t))]N

i,k=1 = α(t), where α(t) : [−1,1]→ R1 is con-tinuous function becoming zero only in t = 0.

Theorem 8.1. Let condition A) hold and α(t) is a monotonic increasing function.Then ε0 is bifurcation point of bifurcation equation (8.2.1).

Proof. Let us take arbitrary small r > 0 and δ > 0. Let us consider continuous vectorfield

H(ξ,2)defn= L(ξ,ε((22− 1)δ)) : RN

×R→ RN,

given for ξ,2 ∈M, where M = {ξ,2 | ‖ ξ ‖= r, 0≤2≤ 1}.

Case 1. If there exists a pair (ξ∗,2∗) ∈M with H(ξ∗,2∗)= 0, then by definition 8.2, ε0 isbifurcation point.

Case 2. We assume that H(ξ,2) 6= 0 for ∀(ξ,2) ∈M and, hence, ε0 is not the bifurcationpoint. Then vector fields H(ξ,0) and H(ξ,1) are homotopic on the sphere ‖ ξ ‖= r.Hence, their rotations [161] coincide

J(H(ξ,0),‖ ξ ‖= r)= J(H(ξ,1),‖ ξ ‖= r). (8.2.7)

Since vector fields H(ξ,0), H(ξ,1) and their linerization

L−1 (ξ)defn=

N∑k=1

aik(ε(−δ))ξk |Ni=1,

L+1 (ξ)defn=

N∑k=1

aik(ε(+δ))ξk |Ni=1

are nondegenerated on the sphere ‖ ξ ‖= r, then, due to smallness of r > 0, fields(H(ξ,0), H(ξ,1)) are homotopic in their linear parts L−1 (ξ) and L+1 (ξ).Thus,

J(H(ξ,0),‖ ξ ‖= r)= J(L−1 (ξ),‖ ξ ‖= r), (8.2.8)

J(H(ξ,1),‖ ξ ‖= r)= J(L+1 (ξ),‖ ξ ‖= r). (8.2.9)

Since linear fields L±1 (ξ) are nondegenerate according to the theorem of Kroneckerindex, the following equalities

J(L−1 (ξ),‖ ξ ‖= r)= signα(−δ),

J(L+1 (ξ),‖ ξ ‖= r)= signα(+δ)

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holds. Since α(−δ) < 0, α(+δ) > 0 then, due to (8.2.8), (8.2.9) relation, (8.2.7) is notvalid. Hence, we found a couple (ξ∗,2∗) ∈M, for which H(ξ∗,2∗)= 0 and ε0 isbifurcation point.

Remark 8.1. If conditions of the Theorem 8.1 are satisfied for ∀ε ∈�0 ⊂�, then �0is a bifurcation set of equation (8.2.1). Moreover, if �0 is a connected set and everypoint is contained in their own neighborhood are homeomorphous with some domainfrom RN , then �0 is called n-dimensional bifurcation manifold.

For example, taking ϒ = Rn+1, n≥ 1 we have�0—a bifurcation set of (8.2.1) andit contains the point ε = 0, while ∇ε det[aik(ε)] |ε=0 6= 0.

The generalization of this result (see [272]) follows from Theorem 8.1 with ϒ =R, and also other known strengthenings of Krasnosel’s kij theorem about bifurcationpoint of the odd multiplicity [161].

Stronger results in the theory of bifurcation points are obtained for (8.2.1) with thepotential BEq in ξ , when

L(ξ,ε)= gradξU(ξ,ε). (8.2.10)

This condition is satisfied if matrix

[∂Lk

∂ξi

]N

i,k=1is symmetrical. By means of the dif-

ferentiation of superposition, we find from (8.2.6) that

∂Lk

∂ξi=

⟨Rx

(N∑

s=1

ξsϕs+U(ξ,ε),ε

)(ϕi+

∂U

∂ξi

),ψk

⟩, (8.2.11)

where, according to (8.2.3), (8.2.5),

ϕi+∂U

∂ξi= (I−0Rx)

−1ϕi. (8.2.12)

Operator I−0Rx is continuously invertible, because ‖ 0Rx ‖< 1 for the suffi-ciently small by norm ξ and ε. Substituting (8.2.12) into (8.2.11) we obtain equalities

∂Lk

∂ξi=

⟨Rx(I−0Rx)

−1ϕi, ψk

⟩, i,k = 1, . . . ,n.

The following assertion occurs:

Lemma 8.1. In order for BEq (8.2.2) to be potential, it is sufficient that matrix

4=[⟨

Rx(0Rx)mϕi, ψk

⟩]Ni,k=1

would be symmetrical ∀(x,ε) in the neighborhood of point (0,0).

Corollary 8.1. Let all matrices[⟨Rx(0Rx)

mϕi ,ψk⟩]N

i,k=1 , m= 0,1,2, . . .

be symmetrical in any neighborhood of point (0,0). Then BEq (8.2.2) is potential.

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Corollary 8.2. Let E1 = E2 = H, H—Hilbert space. If operator B is symmetrical inD, and operator Rx(x,ε) is symmetrical for ∀(x,ε) in neighborhood of point (0,0) inD, then BEq is potential.

Work [255] gives more precise conditions for potentiality of BEq.We assume that BEq (8.2.2) is potential. Then it follows from the proof of

Lemma 8.1 that corresponding potential U in (8.2.10) takes the form

U(ξ,ε)=1

2

N∑i,k=1

ai,k(ε)ξiξk+ω(ξ,ε),

where ‖ ω(ξ,ε) ‖= 0(| ξ |2) for ξ → 0.

Theorem 8.2. Let BEq (8.2.2) be potential. Let condition A) hold. Moreover, let sym-metrical matrix [aik(ε(t))] possess ν1 positive eigenvalues for t > 0 and ν2 positiveeigenvalues for t < 0, ν1 6= ν2. Then ε0 will be the bifurcation point of equation (8.2.1).

Proof. Let us take arbitrary small δ > 0 and consider function U(ξ,ε((22− 1)δ)),defined into 2 ∈ [0,1] in neighborhood of critical point ξ = 0.

Case 1. If there exists 2∗ ∈ [0,1] such that ξ = 0 is nonisolated critical point of functionU(ξ,ε((22∗− 1)δ)), then, due to definition 8.2, ε0 is the bifurcation point.

Case 2. We assume that point ξ = 0 is the isolated critical point of function U(ξ,ε((22−1)δ))for ∀2 ∈ [0,1], where ε(t) is continuous function from condition A). Then with ∀2 ∈[0,1] for this function, the Conley index [75] K2 of critical point ξ = 0 is determined.Let us recall that

det ‖∂2U(ξ,ε((22− 1)δ))

∂ξi∂ξk‖ξ=0= α((22− 1)δ).

Since α((22− 1)δ) 6= 0 for 2 6= 12 , then critical point ξ = 0 for 2 6= 1

2 is not degen-

erated. Thus, Conley index K2 with certain 2 6= 12 necessarily is equal to the num-

ber of positive eigenvalues of the corresponding Hessian according to definition(Conley [75], p. 6). Therefore, K2 = ν1, K1 = ν2, where ν1 6= ν2, due to conditionsof Theorem 8.2. Hence, K2 6= K1. We assume that ε0 is not the bifurcation point.Then ∇ξU(ξ,ε((22− 1)δ)) 6= 0 with 0<‖ ξ ‖≤ r, where r > 0 sufficiently smallnumber,2 ∈ [0,1]. In view of the homotopic invariance of Conley index ([75], p. 52,Teorem 4), K2 is a constant for2 ∈ [0,1] and K0 = K1. Hence, in the second case, wewill always find the pair (ξ∗,2∗) satisfying equation ∇ξU(ξ,ε((22− 1)δ))= 0 forarbitrary small r > 0, δ > 0, where 0<‖ ξ∗ ‖≤ r,2∗ ∈ [0,1]. Thus, ε0 is the bifurca-tion point.

Remark 8.2. Another proof of Theorem 8.2 for case ϒ = = R, ν+ = n, ν− = 0 withthe application of the Rolle theorem was given in the paper of Trenogin, Sidorov,Loginov [273].

Remark 8.3. Theorems 8.1, 8.2 (see Remark 8.1) make it possible to build notonly the points of bifurcation, but also bifurcational sets, surfaces and curves ofbifurcation.

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Corollary 8.3. Let ϒ = R and BEq be potential. Furthermore, let [aik(ε)]Ni,k=1 be the

positively defined matrix for ε ∈ (0,r) and negatively defined for ε ∈ (−r,0). Thenε = 0 is the bifurcation point of equation (8.2.1).

Let us consider connection of eigenvalues of matrix [aik(ε)] with eigenvalues ofoperator B−Rx(0,ε).

Lemma 8.2. Let E1 = E2 = E, ε ∈ R; ν = 0 be isolated Fredholm point of operator-function B− νI. Then

sign4(ε)= (−1)ksignk∏i

νi(ε)= signN∏i

µi(ε),

where k is the root number of operator B; {µ}N1 are eigenvalues of matrix [aik(ε)],4(ε)= det[aik(ε)].

Proof. Since {µi}N1 are eigenvalues of matrix [aik(ε)], then

∏Ni µi(ε)=4(ε). Thus,

it suffices to prove equality 4(ε)= (−1)k∏k

i νi(ε). Since zero is isolated Fredholmpoint of operator-function B− νI, then operators B and B∗ have the correspondingcomplete Jordan system [279]

ϕ(s)i = (0)

(s−1)ϕ(1)i , ψ

(s)i = (0

∗)(s−1)ψ(1)i , i= 1, . . . ,n; s= 1, . . . ,Pi.

(8.2.13)

Here

< ϕ(Pi)i ,ψj >= δij; < ϕi,ψ

(Pj)

j >= δij, i, j= 1, . . . ,n;N∑

i=1

Pi = k.

Let us recall that

ϕ(1)i4= ϕi = 0ϕ

(Pi)i , ψ

(1)i4= ψi = 0

∗ψ(Pi)i ,

0 =

(B+

N∑1

< ·,ψ(Pi)i > ϕ

(Pi)i

)−1

, (8.2.14)

where k = l1+ ·· ·+ ln is the root number of operator B−Rx(0,ε). Small eigenvaluesν(ε) of operator B−Rx(0,ε) satisfy the following BEq [279]

L(ν,ε)4= det |<(Rx(0,ε)+ νI)(I−0Rx(0,ε)− ν0)

−1ϕi,ψj>|ni,j=1= 0.

(8.2.15)

Due to a Veyerstrass theorem [279] and relations (8.2.13), (8.2.14), equation (8.2.15)can be transformed into

L(ν,ε)≡ (νk+Hk−1(ε)ν

k−1+ ·· ·+H0(ε))�(ε,ν)= 0

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in the neighborhood of zero, where Hk−1(ε), . . . ,H0(ε)=4(ε) continuous functionsε, �(0,0) 6= 0, H0(0)= 0. Hence, operator B−Rx(0,ε) has k ≥ n small eigenvaluesνi(ε), i= 1, . . . ,n, which we can find from equation

νk+Hk−1(ε)ν

k−1+ ·· ·+4(ε)= 0.

Then∏k

i νi(ε)=4(ε)(−1)k.

Let ε ∈ R. Let us consider computation of asymptotics of eigenvalues µ(ε)and ν(ε). Let us introduce the block presentation of matrix [aik]N

i,k=1 satisfying thefollowing:

Condition (B) Let [aik(ε)]Ni,k=1 = [Aik(ε)]l

i,k=1 ∼ [εrik A0ik]l

i,k=1 for ε→ 0, where

[Aik] blocks of dimension [ni× nk], n1+ ·· ·+ nl = n, min(ri1, . . . ,ril)= rii4= ri and

rik > ri for k > i (or for k < i), i= 1, . . . , l.Let

∏l1 det[A0

ii] 6= 0. Condition B denotes that matrix [aik(ε)]Ni,k=1 admits the block

presentation, which is “asymptotically triangular” for ε→ 0.

Lemma 8.3. Let condition B hold. Then

det[aik(ε)]Ni,k=1 = ε

n1r1+···+nlrl

( l∏1

det | A0ii | +0(1)

),

formulas

µi = εri(Ci+ 0(1)), i= 1, . . . , l, (8.2.16)

define the dominant terms all n eigenvalues of matrix | aik(ε) |Ni,k=1, where µi, Ci ∈

Rni , Ci vector of eigenvalues of matrix A0ii.

Proof. Due to condition B and property of linearity of determinant, we obtain

det[aik(ε)]= εn1r1+···+nlrl det

∣∣∣∣∣∣∣∣A0

11+ 0(1) 0(1) . . . . . . . . . . . .0(1)

A021+ 0(1) A0

22+ 0(1) 0(1) . . .0(1). . . . . . . . . . . . . . . . . . . . . . . . . . .

A0l1+ 0(1) . . . . . . . . . A0

ll+ 0(1)

∣∣∣∣∣∣∣∣== εn1r1+···+nlrl

(l∏i

det |A0ii | + 0(1)

).

Substituting µ= εric(ε), i= 1, . . . , l, into equation det | aik(ε)−µδik |Ni,k=1= 0 and

using the property of linearity of determinant, we obtain the following equation

εn1r1+···+ni−1ri−1+(ni+···+nl)ri

{i−1∏j=1

det | A0jj | ×

× det(A0ii− c(ε)E)c(ε)ni+1+···+nl + ai(ε)

}= 0, i= 1, . . . , l, (8.2.17)

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where ai(ε)→ 0 for ε→ 0. Hence, the coordinates of the dominant terms Ci inasymptotics (8.2.16) satisfy equations det | A0

ii− cE |= 0, i= 1, . . . , l.

If k = n, then operator B−Rx(0,ε), just as matrix [aik(ε)]Ni,k=1 has n small eigen-

values. In this case the following result holds.

Corollary 8.4. Let operator B not have I adjoint elements and assume condition B.Then formula

νi =−εri(Ci+ 0(1)), i= 1, . . . , l, (8.2.18)

defines all n small eigenvalues of operator B−Rx(0,ε), where Ci ∈ Rni is vector ofeigenvalues of matrix A0

ii, i= 1, . . . , l, n1+ ·· ·+ nl = n.

Proof. In this case, due to Lemma 8.2, we have∑N

1 Pi = n (root number k = n) andoperator B−Rx(0,ε) possesses n small eigenvalues. Since

∑l1 ni = n, A0

ii is the squarematrix, then (8.2.18) gives n eigenvalues, where dominant terms coincide with domi-nant terms in (8.2.16) with an accuracy to the sign.

For computing eigenvalues ν of operator B−Rx(0,ε), we transform (8.2.15) into

L(ν,ε)≡ det[aik(ε)+

∞∑j=1

b(j)ik νj]N

i,k=1 = 0, (8.2.19)

where

b(j)ik =< [(I−0Rx(0,ε))−10] j−1(I−0Rx(0,ε))

−1ϕi,γk >.

Substituting ν =−εric(ε) into (8.2.19) and taking into account the property of linear-ity of determinant, we obtain equation which differs from (8.2.17) only in terms oferror of calculation ai(ε).

Then in conditions of Corollary 8.4 the dominant terms define all small eigenvaluesof operator B−Rx(0,ε) and matrices −[aik(ε)] are found from the same equationsand, hence, are identical.

8.3 Conclusions

1. Due to Lemma 8.3, we can change condition A in Theorem 8.1:Condition (A∗) Let E1 = E2 = E, ν = 0 be isolated Fredholm point of operator-function

B− νI. Let in neighborhood of the point ε0 ∈� there be the set S, containing point ε0, whichpresents the continuum S= S+

⋃S−. Furthermore, let us assume that

ε0 ∈ ∂S+⋂∂S−,

∏i

νi(ε)

∣∣∣∣∣ε∈S+

·

∏i

νi(ε)

∣∣∣∣∣ε∈S−

< 0,

where {νi(ε)} are small eigenvalues of operator B−Rx(0,ε).

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2. If the dominant terms in asymptotics of small eigenvalues of operator B−Rx(0,ε) andmatrix [aik(ε)]N

i,k=1 coincide, then Theorem 8.2 is applicable. Due to Corollary 8.4, it is pos-sible, if E1 = E2 = H. Then operators B and Rx(0,ε) are symmetrical and condition B is sat-isfied. We note that condition B is also realized in papers of Sidorov and Trenogin [255, 272]on the bifurcation point with potential BEq, moreover r1 = . . .= rn = 1.

8.4 Statement of Boundary Value Problem and the Problemon Point of Bifurcation of System (8.4.7), (8.4.13)

Let us present one preliminary result about the reduction of VM system (8.1.1)–(8.1.2)with boundary conditions (8.1.3), (8.1.4) to the quasilinear system of elliptic equationsfor the distribution function (8.1.5). Assume that the following condition is satisfied.

Condition (C) fi(R,G) is given, differentiable functions in distribution (8.1.5); αi,di constant parameters; |di| 6= 0, ϕi = c1i+ liϕ(r), ψi = c2i+ kiψ(r), c1i, c2i—Const,parameters li, ki are connected by relations

li =m1

α1q1

αiqi

mi, ki

q1

m1d1 =

qi

midi, k1 = l1 = 1, (8.4.1)

and integrals∫R3

fidv,∫R3

fivdv converge for ∀ϕi, ψi.

Let us introduce the following notations m14= m, α1

4= α, q1

4= q.

Theorem 8.3. Let fi be defined in (8.1.5) and condition C hold. Also assume thatvector-function (ϕ,ψ) is a solution of the system of equations

4ϕ = µ

N∑k=1

qk

∫R3

fkdv, µ=8παq

m,

4ψ = ν

N∑k=1

qk

∫R3

(v,d)fkdv, ν =−4πq

mc2, (8.4.2)

ϕ |∂D =−2αq

mu01, ψ |∂D=

q

mcu02

in subspace

(∂rϕi,di)= 0, (∂rψi,di)= 0, i= 1, . . . ,N. (8.4.3)

Then VM system (8.1.1)–(8.1.4) has a solution

E =m

2αq∂rϕ, B=

d

d2

β + 1∫0

(d× J(tr),r)dt

− [d× ∂rψ]mc

qd2, (8.4.4)

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where

J4=

4π

c

N∑k=1

qk

∫R3

vfkdv, β −Const.

Potentials

U =−m

2αqϕ, A=

mc

qd2ψd+A1(r), (A1,d)= 0, (8.4.5)

satisfying condition (8.1.3), are defined over this solution.

We introduce the notations

ji =∫R3

vfidv, ρi =

∫R3

fidv, i= 1, . . . ,N

and introduce the following condition:Condition (D) there exist vectors βi ∈ R3 such that ji = βiρi, i= 1, . . . ,N.For example condition D is satisfied for distribution

fi = fi(a(−αiv2+ϕi)+ b((di,v)+ψi)) (8.4.6)

for βi =b

2αiadi, a,b—Const.

We assume that condition D holds. Then system (8.4.2) is transformed into

4ϕ = λµ

N∑i=1

qiAi, 4ψ = λν

N∑i=1

qi(βi,d)Ai, (8.4.7)

where

Ai(liϕ,kiψ,αi,di)4=

∫R3

fidv, (8.4.8)

βi =b

2αiadi, (βi,d)=

b

a

d2

2α

ki

li,

(di,d)

αi=

d2

α

ki

li.

In the case of normalized distribution functions, this system admits the followinggeneralization. Let∫

D

∫R3

fidvdr = Ni,

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where

fi = f1i(ϕ,ψ,v)+ f2i(ϕ,ψ) ·M, (8.4.9)

and the integral∫

R3 M(v)dv converges. Then one has the integral identity

∫R3

fidv=∫R3

f1i(ϕ,ψ,v)dv+ f2i(ϕ,ψ) ·

(Ni−

∫D

∫R3 f1idvdr

)∫

D f2idr.

Therefore, for the functions ϕ and ψ occuring in the distribution fi in the case ofnormalized distributions functions fi, we obtain the following system of quasilinearintegro-differential equations

4ϕ = λµ

N∑i=1

qiAi, 4ψ = λν

N∑i=1

qi(βi,d)Ai, (8.4.10)

Ai =

∫R3

f1idv+ λf2i ·

(ξi−

1λ

∫D

∫R3 f1idvdr

)∫

D f2idr,

where λ= ||N||, N4= (N1, . . . ,Nn), ξi = Ni/||N||.

Remark 8.4. Apparently distribution functions of the form (8.4.9) can be useful forthe analysis of stationary solutions of the Boltzmann equation, since they permit oneto simplify the collision integral by separating the variables r and v. If, in addition, wenormalize the function f1i by∫

D

∫R3

f1idvdr = Ki,

then we can study distribution functions with different numbers Ni and Ki.

From now on, for simplicity, we consider the auxiliary vector d in (8.1.5) directedalong the axis Z. Due to (8.4.3), we can take ϕ = ϕ(x,y), ψ = ψ(x,y), x,y ∈ D⊂ R2

in system (8.4.7). Moreover let N ≥ 3 and kili6= Const.

Let D—be a bounded domain in R2 with boundary ∂D of class C2,α ,α ∈ (0,1). Boundary conditions (8.1.3), (8.1.4) for local densities of charge and cur-rent provide equalities:

N∑k=1

qkAk(lkϕ0,kkψ

0,αi,di)= 0;

N∑k=1

qk(βk,d)Ak(lkϕ0,kkψ

0,αi,di)= 0 (8.4.11)

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for ∀ε ∈ ι, where ι is a neighborhood of the point ε = 0 and

ϕ0=−

2αq

mu01, ψ0

=q

mcu02. (8.4.12)

Remark 8.5. If N = 2 and βi =di

2αi, then, due to (8.4.11), (8.4.12) and (βi,d)=

d2

2αkili

,we obtain altarnative: either in (8.4.11) A1 = A2 = 0 or ki = li, i= 1,2. In this case,and also for ki

li= Const, system (8.4.7) is reduced to one equation and the bifurcation

of the solutions is impossible.

Using (8.4.11), (8.4.12) for the system (8.4.7) with boundary conditions

ϕ |∂D= ϕ0, ψ |∂D= ψ

0 (8.4.13)

one obtains a trivial solution ϕ = ϕ0, ψ = ψ0, ∀λ ∈ R+.Then, due to Theorem 8.3 for any λ, the VM system with boundary conditions

(8.1.3), (8.1.4) has the trivial solution

E 0=

m

2αq∂rϕ

0= 0, B 0

= βd1, r ∈ D⊂ R2,

f 0= λfi(−αiv

2+ c1i+ liϕ

0, (v,di)+ c2i+ kiψ0).

Under this condition, ρ and j vanish in domain D.Thus, our aim is to find λ0 providing nontrivial solution for neighborhood systems

(8.4.7), (8.4.13). Then the corresponding densities ρ and j vanish in domain D, andpoint λ0 is the bifurcation point of VM system (8.4.1), (8.4.2), (8.1.3), (8.1.4).

Let functions fi—analytical in (8.1.5). Using the Taylor series expansion

A(x,y)=∞∑

i≥0

1

i!

((x− x0)

∂

∂x+ (y− y0)

∂

∂y

)i

A(x0,y0)

and expressing the linear terms, we can rewrite system (8.4.7) in operator form

(L0− λL1)u− λr(u)= 0. (8.4.14)

Here

L0 =

[4 00 4

], u= (ϕ−ϕ0, ψ −ψ0)′; (8.4.15)

L1 =

N∑s=1

qs

[µls

∂As∂x µks

∂As∂y

νls(βs,d)∂As∂x νks(βs,d)

∂As∂y

]x=lsϕ0,y=ksψ0

4=

4=

[µT1 µT2νT3 νT4

], (8.4.16)

r(u)=∞∑i≥l

N∑s=1

%is(u)bs, (8.4.17)

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where

%is(u)4=

qs

i!

(lsu1

∂

∂x+ ksu2

∂

∂y

)i

As(lsϕ0,ksψ

0)

are ith-order homogeneous forms in u;

∂ i1+i2

∂xi1∂yi2As(x,y)

∣∣∣∣x=lsϕ0, y=ksψ0

= 0

with

2≤ i1+ i2 ≤ l− 1, s= 1, . . . ,N, l≥ 2, bs4= (µ,ν(βs,d))

′.

The existence problem for a bifurcation point, λ0, of the system (8.4.7), (8.4.13)also can be stated as the existence problem for a bifurcation point for the operatorequation (8.4.14).

Let us introduce the Banach spaces C2,α(D) and C0,α(D) with the norms ‖ · ‖2,α ,‖ · ‖0,α , respectively, and let W2,2(D) be the ordinary Sobolev L2 space in D.

Let us introduce the Banach space E of vectors u4= (u1,u2)

′, where ui ∈ L2(D), L2 isthe real Hilbert space with inner product (·, ·) and the corresponding norm ‖ · ‖L2 (D).

We define the domain D(L0) as the set of vectors u4= (u1,u2) with ui ∈

◦

W 2,2(D). Here◦

W 2,2(D) consists of W2,2 functions with zero trace on ∂D. Hence, L0 : D⊂ E→ E isa linear self-adjoint operator. By virtue of the embedding,

W2,2(D)⊂ C0,α(D), 0< α < 1, (8.4.18)

the operator r : W2,2⊂ E→ E—is analytic in a neighborhood of the origin. The oper-

ator L1 ∈ L (E→ E) is linear, bounded. We keep same notations for matrix corre-sponding to operator L1. By the embedding (8.4.18) any solution of equation (8.4.14)is a Holder function in D(L0). Moreover, since the coefficients of system (8.4.14) areconstant, the vector r(u) is analytic, and ∂D ∈ C2,α; it follows from well known resultsof regularity theory for weak solutions of elliptic equations (see Ladyzhenskaya and

Ural’zeva [168]) that the generalized solutions of equation (8.4.14) in◦

W 2,2(D) actu-ally belong to C2,α .

Definition 8.3. (See [274]). A point λ0 is called a bifurcation point of the prob-lem (8.4.7), (8.4.13) if every neighborhood of the point (ϕ0,ψ0,λ0) contains a point(ϕ,ψ,λ) satisfying system (8.4.7), (8.4.13) such that

||ϕ−ϕ0|| ◦

W2,2+ ||ψ −ψ0

|| ◦W2,2

> 0.

Here || · || ◦W2,2

—is the norm in the space W2,2(D).

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According to the Theorem 8.3 on the reduction of the VM system, the bifurcationpoints of problem (8.4.8), (8.4.13) will be reffered to as the bifurcation points of thestationary VM system (8.4.1), (8.4.2), (8.1.3), (8.1.4).

Under the above assumptions about L0 and L1 all singular points of the operator

L(λ)4= L0− λL1 are Fredholm points. If N(L(λ0))= {0} then by the implicit operator

theorem [274], for any δ > 0 there exists a neighborhood S of the point λ0 such thatfor all λ ∈ S the ball ||u||E < δ contains only the trivial solution u= 0, so that λ0 isnot a bifurcation point. Therefore, to find the bifurcation points it is necessary (but notsufficient) to find number λ0 such that N(L0− λ0L1) 6= {0}.

The bifurcation points of the nonlinear equation (8.4.14) are necessarily spectralpoints of the linearized system

(L0− λL1)u= 0. (8.4.19)

To analyze the spectral problem (8.4.19) for physically acceptable parameter val-ues, first we search for eigenvalues and eigenvectors of the matrix L1 in (8.4.19). Toachieve it, we need several auxiliary assertions to be made.

Let us introduce the conditions:

(i) T1 < 0,(ii) (T1T4−T2T3) > 0.

Lemma 8.4. If

∂ fk∂x|x=lkϕ0 > 0,

then condition (i) is satisfied.

Proof. We can assume that q4= q1 < 0, qi > 0, i= 2, . . . ,N, and signqili = signq. By

the assumption of the lemma we have

∂Ai

∂x=

∫R3

∂ fi∂x

dv> 0.

Therefore T1 < 0.

Let us introduce matrix2= ||2ij||i,j=1,...,n 6= [0],2ij = qiqj(ljki− kjli)(βi−βj,d).

Lemma 8.5. If

∂Ai

∂x=∂Ai

∂y, i= 1, . . . ,N, N ≥ 3,

∂Ai

∂x> 0

and the matrix 2—is positive, then conditions (i) and (ii) are satisfied.

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Proof. For any ∂Ai/∂x, we have the identity

T1T4−T2T3 =∑

liai

∑ki(βi,d)ai−

∑kiai

∑li(βi,d)ai =

=

N∑i=2

i−1∑j=1

aiaj(ljki− kjli)(βi−βj,d), where ai4= qi

∂Ai

∂x.

By virtue of this identity, for the expression T1T4−T2T3 to be positive it is sufficientthat

2ij = qiqj(ljki− kjli)(βi−βj,d) > 0, i, j= 1, . . . ,N, N ≥ 3. (8.4.20)

Since the matrix 2—is positive we have (8.4.20).

Remark 8.6. If βi = di/(2αi), then

(βi,d)=d2

2α

ki

li(8.4.21)

and

2ij =d2

2α

qiqj

lilj(ljki− likj)

2 > 0, i, j= 1, . . . ,n,

because sign(qi/li)= signq.

Remark 8.7. If N = 2 and βi = di/(2αi) then by conditions i), ii) and (8.4.21), the fol-lowing alternative takes place: either A1 = A2 = 0 under conditions (8.4.11), (8.4.12)or ki = li, i= 1,2.

For example if βi =di

2αi, then (βi,d)=

d2

2αkili

and

N∑i=2

i−1∑j=1

= aiaj(ljki− likj)2·

d2

2αlilj> 0.

Lemma 8.6. Let distribution function have the form (8.1.5) and f ′i > 0. Then condi-

tions D and (i), (ii) are satisfied for βi =ba

di2αi

, and system (8.4.7) is transformed topotential type

4

[ϕ

ψ

]= λ

[a1 00 a2

][ ∂V∂ϕ∂V∂ψ

], (8.4.22)

where

V =N∑

k=1

qk

lk

alkϕ+bkkψ∫0

Ak(s)ds, a1 = µ/a, a2 =νd2

2ab. (8.4.23)

To prove it we just substitute (8.4.23) into (8.4.22).

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Lemma 8.7. Let r4= x ∈ R, v ∈ R2, d

4= d2. Then system (8.4.22) with potential

(8.4.23) can be written as Hamiltonian system

pϕ =−∂ϕH, ϕ = ∂pϕH,

pψ =−∂ψH, ψ = ∂pψH

with Hamiltonian

H =−p2ϕ

2−

p2ψ

2+V(ϕ(x),ψ(x)).

Here

V(ϕ,ψ)= λa1

N∑k=1

qk

lk

alkϕ∫0

∫R2

A(s,ψ)ds+ λa2

N∑k=1

qk

lk

bkkψ∫0

∫R2

A(ϕ,s)ds.

Proof follows from Lemma 2.2 (see Guo, Ragazzo [126], p. 1152).

Lemma 8.8. Let conditions (i), (ii) be satisfied. Then the matrix L1 in (8.4.16) has onepositive eigenvalue

χ+ = µT1+ 0(1)

and one negative eigenvalue

χ− = ηT1T4−T2T3

T1ε+O(ε), η =

4π | q |

m> 0 (8.4.24)

as ε4=

1c2 → 0.

Eigenvalue χ− generate eigenvectors of matrices L1 and L′1, respectively,[c1c2

]=

[−

T2T1

0

]+O(ε),

[c∗1c∗2

]=

[01

]+O(ε).

Proof. The characteristic equation of the matrix[µT1−χ µT2+εηT3 +εηT4−χ

]has the form

χ2−χ(µT1+ εηT4)+ εηµ(T1T4−T2T3)= 0.

Since

χ1,2 =1

2

(µT1+ εηT4±

√(µT1+ εηT4)2− 4εηµ(T1T4−T2T3)

),

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we obtain

χ+ = µT1+O(1), χ− = εηT1T4−T2T3

T1+O(ε)

as ε→ 0. Since µ < 0 and T1 < 0 it follows that χ+ > 0. It can be checked in a similarway that χ− < 0 by virtue of the inequalities η > 0, T1T4−T2T3 > 0, T1 < 0.

Solving the homogeneous systems

(4−χ−)c= 0, (4′−χ−)c∗= 0,

we can find the eigenvectors corresponding to χ−.

Let us proceed to the computation of the bifurcation point λ0. Assuming that λ=λ0+ ε in (8.4.14), consider the system

(L0− (λ0+ ε)L1)u− (λ0+ ε)r(u)= 0 (8.4.25)

in a neighborhood of the point λ0. Assume that either T2 6= 0 and T3 6= 0 or T2 =

T3 = 0. To symmetrize the system for T2 6= 0, T3 6= 0, we multiply both sides ofequation (8.4.25) by matrix

M =

(1 00 a

), where a

4=µT2

νT36= 0.

We rewrite (8.4.25) in the form

Bu= εB1u+ (λ0+ ε)R(u). (8.4.26)

Here

B=M(L0− λ0L1), R(u)4=Mr(u)

4= (r1(u),r2(u)),

B1 ∈ L(E→ E)—is a self-adjoint operator, since it is generated by the symmetricmatrix B1 =ML1, and B : D(L0)⊂ E→ E—is a self-adjoint Fredholm operator.

Remark 8.8. If As(alsϕ+ bksψ) then

∂As

∂y= A′sb,

∂A

∂x= A′sa, a=

µb

ν

d2

2αa, βs =

b

a

ds

2αs.

In decomposition (8.4.17)

%is =qs

i!A(i)s (alsϕ

0+ bksψ

0)(alsu1+ bksu2)i.

Therefore, ∂r1/∂u2 = ∂r2/∂u1 in this case, the matrix Ru(u) is symmetric for any u,and the operator Ru : E→ E—is self-adjoint for any u.

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Remark 8.9. If T2 = T3 = 0 then one can set a= 1. If either T2 = 0 and T3 6= 0 orT3 = 0 and T2 6= 0 then the problem cannot be symmetrized and the derivation of theBEq in Subsection 8.5 must be performed directly for equation (8.4.25).

Let us introduce the following condition:(iii) Let µ—be an eigenvalue of the Dirichlet problem

−4e= µe, e|∂D = 0,

and {e1, . . . ,en}—be an orthonormal basis in the subspace of eigenfunctions.Let c− = (c1,c2)

′—be the eigenvector of the matrix L1 corresponding to the eigen-value χ− < 0.

Lemma 8.9. Let condition (iii) hold, and λ0 =−µ/χ−. Then dimN(B)= n and thesystem {ei}

Ni=1, where ei = c−ei is a basis in the subspace N(B).

Proof. Consider the matrix 0 whose columns are the eigenvectors of L1 correspond-ing to the eigenvalues χ−, χ+. Then

0−1L10=

(χ− 00 χ+

), L00= 0L0,

and substitution u= 0U reduces the equation Bu= 0 into

M[L00U− λ0L10U]=M[0(L0U− λ00−1L10U)]= 0.

It follows that the linearized system (8.4.19) splits into the two linear elliptic equations

4U1− λ0χ−U1 = 0, u1|∂D = 0, (8.4.27)

4U2− λ0χ+U2 = 0, u2|∂D = 0, (8.4.28)

where λ0χ− =−µ, λ0χ+ > 0. By condition (iii) we have µ ∈ σ(−4). Therefore,

U1 =

N∑i=1

αiei, αi = Const, U2 = 0,

and, hence,

∣∣∣∣ u1u2

∣∣∣∣= 0U =

∣∣∣∣ c1− c1+c2− c2+

∣∣∣∣ · ∣∣∣∣u10

∣∣∣∣= ∣∣∣∣ c1−c2−

∣∣∣∣ N∑i=1

αiei.

Lemma 8.10. The operator B does not have B1-adjoint elements.

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Proof. Since L1c− = χ−c−, we obtain an equility

< B1ei,ek >=

⟨∣∣∣∣ 1 00 a

∣∣∣∣L1c−ei,c−ek

⟩= χ−

⟨∣∣∣∣ c1−c2−

∣∣∣∣ei,

∣∣∣∣ c1−c2−

∣∣∣∣ek

⟩=

= χ−(c21−+ ac2

2)δik, i,k = 1, . . . ,n.

Therefore, det||< B1ei,ek > ||Ni,k=1 = χ

N−(c

21−+ ac2

2−)N6= 0, because

χ− 6= 0, c21i+ ac2

2− ≈−2αT2

T3c2,

and, hence, according to the definition of generalized Jordan sequences [279], theoperator B does not have B1-adjoint elements.

Without loss of generality we can assume that the eigenvector c1− of the matrixL1 is chosen so that χ−(c2

1−+ ac22−)= 1. Then the system of vectors {B1ei}

Ni=1 is

biorthogonal to {ei}Ni=1. Hence, by Schmidt’s Lemma [279] the operator

B= B+N∑

i=1

< ·,γi > γi,

with γi4= B1ei, has a bounded inverse 0 ∈ L (E→ E). Thus,

0 = 0∗, 0γi = ei. (8.4.29)

Remark 8.10. It follows from the proof of Lemma 8.7 that to construct the operator0 one can use the equation

0 = 0

∣∣∣∣01 00 02

∣∣∣∣0−1M−1,

where

01 =

∫D

G1(x,s)[·]ds, 02 =

∫D

G2(x,s)[·]ds,

G1(x,s)—is the modified Green’s function of the Dirichlet problem (8.4.27), andG2(x,s)—is the Green’s function of the Dirichlet problem (8.4.28).

8.5 Resolving Branching Equation

Let us rewrite equation (8.4.26) in the form of the system

(B− εB1)u= (λ0+ ε)R(u)+∑

i

ξiγi, (8.5.1)

ξi =< u,γi >, i= 1, . . . ,n. (8.5.2)

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From (8.5.1), by the inverse operator theorem, we have

u= (λ0+ ε)(I− ε0B1)−10R(u)+

1

1− ε

N∑i=1

ξiei. (8.5.3)

Moreover, in virtue of (8.5.2) and (8.4.29) we must have

ε

1− εξi+

λ0+ ε

1− ε< R(u),ei >= 0, (8.5.4)

where R(u)= Rl(u)+Rl+1(u)+ ·· · . According to the implicit operator theoremequation (8.5.3) has the unique solution

u= u1(ξe,ε)+ (λ0+ ε)(I− ε0B1)−10{ul(ξe,ε)+ ul+1(ξe,ε)+ . . .} (8.5.5)

for sufficiently small ε and |ξ |, where

u1(ξe,ε)=1

1− ε

N∑i=1

ξiei, ul(ξe,ε)= Rl(u1(ξe,ε)),

ul+1(ξe,ε)= Rl+1(u1(ξe,ε))+

+

{0, l> 20R

′

2(u1(ξe,ε))(λ0+ ε)(I− ε0B1)−10u2(ξe,ε), l= 2,

and so on.By substituting the solution (8.5.5) into (8.5.2), we obtain the desired bifurcation

system (BEq)

ε

1− εξ +L(ξ,ε)= 0, (8.5.6)

where L= (L1, . . . ,LN),

Li =λ0+ ε

(1− ε)l+1

[⟨Rl(ξe),ei

⟩+

1

1− ε

⟨Rl+1(ξe),ei

⟩]+

+

0, l> 2λ0+ε

(1−ε)4

⟨R′2(ξe)(I− ε0B1)

−10R2(ξe),ei

⟩, l= 2

+ ri(ξ,ε),

ri = o(|ξ |l+1), i= 1, . . . ,n.

If L(ξ,ε)= gradU(ξ,ε), then the BEq (8.5.6) is said to be potential. In the poten-tial case matrix Lξ (ξ,ε) is symmetric. Let fi = fi(aliϕ+ bkiψ), i= 1, . . . ,N in (8.1.5).Then by Remark 8.8 the matrices Ru(u) are symmetric for any u.

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Let us show that the BEq (8.5.6) is potential if the matrix u(u)—is symmetric forany u. Indeed, by (8.5.4)

Li(ξ,ε)=λ0+ ε

(1− ε)

⟨R(u(ξe,ε)),ei

⟩, i= 1, . . . ,n,

where u(ξe,ε) is defined by the series (8.5.5).Therefore, the vector field L(ξ,ε) is potential if and only if the matrix

∣∣∣∣∣∣∣∣⟨Ru∂u(η,ε)

∂ηej,ei

⟩∣∣∣∣∣∣∣∣Ni,j=1

(η4= ξe) (8.5.7)

is symmetric.By virtue of (8.5.3) and the inverse operator theorem we have the operator identity

∂u(η,ε)

∂η=

[I− (λ0+ ε)(I− ε0B1)

−10Ru(u(η,ε))

]−1 1

1− ε

in a sufficiently small neighborhood of the point ξ = 0, ε = 0. Since B1, 0 and Ru—are self-adjoint operators it follows that

[∂u

∂η

]∗= [I− (λ0+ ε)Ru(u(η,ε))]

−1 1

1− ε.

Therefore,[Ru∂u

∂η

]∗= [I− (λ0+ ε)Ru0(I− εB10)

−1]−1Ru1

1− ε= Ru

∂u

∂η

by virtue of the operator identity

[I− (λ0+ ε)Ru0(I− εB10)−1]−1Ru = Ru[I− (λ0+ ε)(I− ε0B1)

−10Ru]−1.

Hence, the operator

Ru∂u

∂η: E→ E

in the matrix (8.5.7) is self-adjoint. Therefore, the matrix (8.5.7) is symmetric and

L(ξ,ε)= gradU(ξ,ε).

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The foregoing implies the following assertion:

Lemma 8.11. Let conditions (8.4.11), (8.4.12), (i)–(iii) be satisfied and λ0 =−µ/χ−.Then the number of solutions of Eq. (8.4.25) such that u→ 0 as for λ→ λ0 coincideswith the number of small solutions ξ → 0 as ε→ 0 of the BEq (8.5.6).

If Ai = Ai(aliϕ+ bkiψ), i= 1, . . . ,N in system (8.4.2), (8.4.3), (8.4.7), (8.4.13), anda, b—are constants, then the BEq is potential, that is, L(ξ,ε)=∇ξU(ξ,ε), U : RN

×

R→ R, where

u(ξ,ε)=−λ0+ ε

(l+ 1)(1− ε)l+1

N∑i=1

⟨Rl(ξe),ei

⟩ξi−

−λ0+ ε

(l+ 2)(1− ε)l+2

N∑i=1

⟨Rl+1(ξe),ei

⟩ξi−

−

0, l> 2

(λ0+ ε)(1−ε)4

N∑i=1

⟨R′2(ξe)(I− ε0B1)

−10R2(ξe),ei

⟩ξi, l= 2

++ o(|ξ |l+1). (8.5.8)

8.6 The Existence Theorem for Bifurcation Points andthe Construction of Asymptotic Solutions

The BEq (8.5.6) is the desired Lyapunov-Schmidt BEq (see Vainberg andTrenogin [279]) for the bifurcation point of the boundary value problem (8.4.7),(8.4.13). In the sequel we need some properties of the real solutions and the struc-ture of the BEq

L(ξ,ε)4=

ε

1− εξ +Ll(ξ,ε)+ o(|ξ |l)= 0. (8.6.1)

We state these results from [255, 272] in the form of two lemmas.

Lemma 8.12. Let

1. n be odd; or2. l be even and

∑Nj=1 |L

jl(ξ,0)| 6= 0 for ξ 6= 0 or

3. L(ξ,ε)=∇W(ξ,ε).

Then in every neighborhood of the point ξ = 0,ε = 0 there exists a pair (ξ∗,ε∗), ξ∗ 6=0 satisfying (8.6.1).

Proof. If the point ξ = 0 is a nonisolated singular point of the vector field L(ξ,0),then in every neighborhood of the point ξ = 0, ε = 0 there exists a pair (ξ∗,0) suchthat ξ∗ 6= 0 satisfies (8.6.1), and the lemma is true.

Assume ξ = 0—is an isolated point of the vector field L(ξ,0). Let us consider threecases:

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1. Let n be odd. Let us take the neighborhood |ξ | ≤ r, |ε| ≤ % and introduce the vector field

8(ξ, t)=(2t− 1)%

1− (2t− 1)%ξ +L(ξ, (2t− 1)%).

If 8(ξ, t) 6= 0 for t ∈ [0,1] and |ξ | = r then the degree of the map

Jt = J

(8

||8||,S(0,r)

)of the boundary of the sphere |ξ | = r into the unit sphere is well defined [12], and, hence,Jt—is the same integer for each t ∈ [0,1]. But J0 = (−1)N , J1 = 1N . Hence, Jt 6= Const.Therefore, for all r > 0 and % > 0 there exist t∗ ∈ [0,1] and ξ∗, |ξ∗| = r such that8(ξ∗, t∗)= 0. The corresponding pair (ξ∗, (2t∗− 1)%) satisfies system (8.6.1).

2. Let l be even and let

N∑j=1

|Ljl(ξ,0)| 6= 0.

In this case, the field L(ξ,ε) is homotopic to Ll(ξ,0) on the sphere S(0,r) for |ε|< δ with δsufficiently small. Hence,

J

(L(ξ,ε)

||L||,S(0,r)

)= J

(Ll(ξ,0)

||Ll||,S(0,r)

)is an even number, because l is even.

Let us fix an ε∗ ∈ (−δ,δ) and introduce the Kronecker index [163] γ0 of the isolatedsingular point ξ = 0 of the field L(ξ,ε∗), γ0 = (signε∗)N .

By Kronecker theorem [163]

J

(L(ξ,ε∗)

||L||,S(0,r)

)=

∑i

γi. (8.6.2)

Since the left-hand side of equation (8.6.2) is even, and γ0—is odd, it follows that along withthe point ξ = 0 the sphere S(0,r) contains another singular point ξ∗ 6= 0 of the field L(ξ,ε∗).The pair (ξ∗,ε∗) satisfies system (8.6.1).

3. Let L(ξ,ε)= gradW(ξ,ε), where

W(ξ,ε)=ε

2(1− ε)

∑i

ξ2i +U(ξ,ε), |ξ |< δ, δ > 0.

If the point ξ = 0 is a nonisolated critical point of the potential W(ξ,0) (of the poten-tial W(ξ,ε) for 0< |ε|< δ) then the lemma is true. Let the point ξ = 0—be an isolatedcritical point of the potential W(ξ,0) and of the potential W(ξ,ε) for 0< |ε|< δ. Thenthe Morse-Conley indices (see Conley [75]) are defined for the critical point ξ = 0 of thepotential W(ξ,0) and of the potential W(ξ,ε) for 0< |ε|< δ, where δ is sufficiently small.By the homotopic invariance property of the Morse-Conley index (see Conley [75], p. 67,Theorem 1.4), these indices are equal. But for 0< |ε|< δ the point ξ = 0 is a nondegeneratecritical point, because

det

∣∣∣∣∣∣∣∣ ∂2

∂ξi∂ξjw(ξ,ε)

∣∣∣∣∣∣∣∣ε=0=

εN

(1− ε)N6= 0.

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Therefore, the Morse-Conley index is equal to the number of negative eigenvalues of thecorresponding Hessian∣∣∣∣∣∣∣∣ ε

1− εδik

∣∣∣∣∣∣∣∣i,k=1,...,n

.

But then this index is zero for ε > 0, and for ε < 0 it is equal to n. Therefore, the point ξ = 0cannot be an isolated critical point of the potentials W(ξ,0) and W(ξ,ε) for 0< |ε|< δ.Hence, ξ = 0 is a bifurcation point of the BEq L(ξ,ε)= 0.

Lemma 8.13. 1. Let l—be even, and let the system

ξ +Ll(ξ,0)= 0 (8.6.3)

have a simple real solution ξ06= 0, then in a neighborhood of the point ε = 0, system (8.6.1)

has a real valued solution of the form

ξ = (ξ0+ o(1))ε1/(l−1). (8.6.4)

2. Let l—be odd, and let system (8.6.3) or the system

−ξ +Ll(ξ,0)= 0 (8.6.5)

have a simple real solution ξ06= 0, then in the half-neighborhood ε > 0 (ε < 0) there exist

two solutions of the form

ξ = (±ξ0+ o(1))|ε|1/(l−1). (8.6.6)

3. Let Ll(ξ,0)= gradU(ξ) and let ξ0—be an isolated extremum of the function U(ξ) on thesphere |ξ | = 1, U(ξ0) 6= 0, then there exists a solution of the form

ξ = (c+ o(1))|ε|1/(l−1), (8.6.7)

where

c=

(

signε

(l+ 1)U(ξ0)

)1/(l−1)

ξ0,

±

(1

(l+ 1)U(ξ0)

)1/(l−1)

ξ0, εU(ξ0) > 0.

(8.6.8)

Proof. 1), 2). We seek the solutions of equation (8.6.3) in the form

ξ = η(ε)ε1/(l−1).

To define η(0) we obtain two systems: one for ε > 0 and the other for ε < 0, that, aresystems (8.6.3), (8.6.5). If l—is even then the substitution ξ =−ξ transforms equa-tion (8.6.3) into equation (8.6.5). If l—is odd this substitution does not change equa-tions. Therefore, in the case of simple real solutions ξ0 the existence of solutions ofthe form (8.6.4), (8.6.6) follows from the implicit function theorem.

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3). Let Ll(ξ,0)= gradU(ξ). Then we seek the solutions of (8.6.7) in the form

ξ =

(ε

2q

)1/(l−1)

ε,

where |η| = 1 and q(ε)—is a scalar parameter satisfying the condition εq(0) > 0 forodd l. For q and η we obtain the system

2qη+Ll(η,0)+2(η,q,ε)= 0, |η| = 1,

where ||2|| = o(1) as ε→ 0. Therefore, 2q(0)η(0)+Ll(η(0),0)= 0 and |η(0)| = 1.Since, by assumption, ξ0—is an isolated extremum of the function U(ξ) on the

sphere |ξ | = 1 and U(ξ0) 6= 0, we set

q(0)= (l+ 1)U(ξ0), η(0)= ξ0.

Consider the perturbed vector field

8ε(η,q)=

{2qηi+

∂U

∂ηi+2i, |η| = 1

}.

Let S—be the sphere of radius % > 0 centered at the point (q(0),η0) in Rn+1. Let usintroduce the degree of the map (see Rothe [258]):

J

(8ε

||8ε ||,S

)=

=

{+1, if (q(0),ξ0)—is an arg of the min q(0)|ξ |2+U(ξ,0),(−1)n+1, if (q(0),ξ0)—is an arg of the max q(0)|ξ |2+U(ξ,0).

Since this degree is nonzero, it follows that the vector field8ε(η,q)= 0 has a singularpoint in a neighborhood of the point (q(0),ξ0) for |ε|< δ with δ sufficiently small.

With the help of Lemmas 8.11, 8.12, and 8.13, it is now possible to prove thefollowing results on the bifurcation point for problems (8.4.7), (8.4.13).

Theorem 8.4. Let conditions (8.4.11), (8.4.12), (i)–(iii) with λ0 =−µ/χ−, as well asone of the following three conditions, be satisfied:

1. n is odd;2. l is even and

∑Ni=1 |< Rl(ξe),ei > | 6= 0 for ξ 6= 0;

3. fi = fi(a(−αiv2+ϕi)+ b(vdi+ψi)), i= 1, . . . ,N, and a, b—are constants.

Then λ0—is a bifurcation point of the boundary value problem (8.4.7), (8.4.13).

Proof. By assumptions (1)–(3) of the theorem the assumptions of Lemmas 8.11 and8.12 are satisfied for the BEq (8.5.6) of the boundary value problem (8.4.7), (8.4.13).Equation (8.5.5) establishes a one-to-one correspondence between the desired solu-tions of the boundary value problem and small solutions of the BEq (8.5.6). Therefore,the validity of Theorem 8.4 follows from these lemmas.

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Corollary 8.5. Let the potentials of the electromagnetic field satisfy conditions (8.1.3),and let the assumptions of Theorem 8.4 be satisfied. Then λ0—is a bifurcation point ofthe VM system (8.1.1)–(8.1.4).

Example 8.1. Let the distribution functions of the VM system have the form [248]

fi = λexp(−αiv2+ (di,v)+ γi+ liϕ(r)+ kiψ(r)),

and

N∑i=1

qiα−3/2i exp

(γi+

d2i

4αi

)= 0,

N∑i=1

qiα−3/2i exp

(γi+

d2i

4αi

)ki

li= 0.

Then conditions (8.4.11), (8.4.12) for βi = di/(2αi) and assumptions (1)–(3) ofTheorem 8.4 are satisfied. Thus, the BEq (8.5.6) is potential. If µ—is an eigenvalue ofthe Dirichlet problem (iii) then, by Corollary 8.5, λ0 =−µ/χ− is a bifurcation pointof the VM system with conditions (8.1.3), (8.1.4), where u10 = u20 = 0. Thus,

T14=

∑aili, T2

4=

∑aiki, T3

4=

∑aili,

T44=

∑ai

k2i

li, ai

4= qi

(π

αi

)3/2

exp

(γi+

d2i

4αi

),

χ− =1

2

(νT4−µT1−

√(νT4−µT1)2+ 4νµ[T1T4−T2T3]

).

Theorem 8.5. Let conditions (8.4.11), (8.4.12), (i)–(iii) be satisfied with λ0 =−µ/χ−as well as the conditions of one of the three statements of Lemma 8.13. If

1) is satisfied then the boundary value problem (8.4.7), (8.4.13) has the solution

u=

(N∑

i=1

ξ0i ei+ o(1)

)(λ− λ0)

1/(l−1);

2) is satisfied then there exist two solutions

u± =

(±

N∑i=1

ξ0i ei+ o(1)

)|λ− λ0|

1/(l−1),

defined in the half-neighborhood λ > λ0 (λ < λ0) provided that ξ0 satisfies system (8.6.3)(ξ0 satisfies system (8.6.5));

3) is satisfied then

u=

(N∑

i=1

ciei+ o(1)

)|λ− λ0|

1/(l−1), (8.6.9)

where the vector c is defined by equation (8.6.8).

The proof follows from Lemmas 8.11, 8.13, and equation (8.5.5).

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Corollary 8.6. Let the potentials of the electromagnetic field satisfy conditions (8.1.3),and let the assumptions of Theorem 8.5 be satisfied. Then the VM system (8.1.1)–(8.1.2) with conditions (8.1.3), (8.1.4) has the solution (8.4.4), where

ϕ =−2αq

mu10+ u1(r,λ), ψ =

q

mcu02+ u2(r,λ), (8.6.10)

the functions u1,u2→ 0 are defined by Theorem 8.5 as λ→ λ0.

Let us consider more detailed distribution functions of the form

fi = fi(a(−αiv2+ c1i+ liϕ)+ b(vdi+ c2i+ kiψ)), (8.6.11)

where li, ki are connected by the linear relations (8.4.1), the integrals∫R3

fidv= Ai(aliϕ+ bkiψ)

converge, and ∂Ai(s)/∂s< 0 for all s. In this case the conditions (8.4.11), (8.4.12),(i), (ii) and the assumptions of Theorem 8.5 are satisfied by Lemmas 8.12, 8.13;hence, according to Lemma 8.12 the BEq (8.5.6) is potential. In Theorem 8.5, case(3) occurs. Therefore, the form of the functions u1(r,λ), u2(r,λ) in (8.6.10) can bespecified; namely, in the case of the distribution (8.6.11) the vector u= (u1,u2) inequation (8.6.10) can be given by equations (8.6.9), (8.6.8). Thus, if it will be foundthat vector c in equation (8.6.8) corresponds to a nonisolated extremum of the cor-responding potential, then some of its coordinates may be arbitrary points of somesphere S⊂ Rk, where k ≤ n (see [255, 272]). Then problem equation (8.4.25) willhave a solution depending on free parameters. This case is possible if the domain D—is symmetric and problem equation (8.4.25) has a spherical symmetry. Thus, the freeparameters remaining in the solution have a group meaning.

Let us show that this is just the situation that arises in our problem in the case of acircular cylinder. Let us introduce the condition

(iv) D= {x ∈ R2|x2

1+ x22 = 1}, and the matrix R′(u)—is symmetric for any u.

Let us pass to the polar coordinates x1ρ cos2, x2 = ρ sin2 in system (8.4.25). Then

4=∂2

∂ρ2+

1

ρ

∂

∂ρ+

1

ρ2

∂2

∂22, u|ρ=1 = 0.

Condition (iii) now makes sense:

µ ∈ {µ(s)2

σ , s= 0,1, . . . , σ = 1,2, . . .},

where µ(s)σ —are the zeros of the Bessel function Js(µ). If µ= µ(s0)2

σ0 , s0 ≥ 1 thendimN(B)= 2; the vectors

c−Js0(µ(s0)σ0ρ)coss02, c−Js0(µ

(s0)σ0ρ)sins02

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form a basis in the subspace N(B). By Lemma 8.11, the BEq (8.5.6) is potential on thewhole. Moreover, the BEq (8.5.6) admits the group O(2) and by [179], Theorem 1,has the form

ξi|ξ |−1L(|ξ |,ε)= 0, (8.6.12)

where

L(|ξ |,ε)=∞∑

i=0

L2i+1(ε)|ξ |2i+1, l1(ε)=

ε

1− ε

is analytical. Let us note that in this case the forms (8.6.10), which are even in ξ , mustbe lacking on the left-hand side of the BEq (8.5.6), since L(|ξ |,ε) is odd in ξ .

Remark 8.11. In view of (8.6.12) the potential U of the BEq in (8.5.8) has the form

U =−

|ξ |∫0

L(s,ε)ds+1

1− ε

ξ21 + ξ

22

2.

Therefore,

U

∣∣∣∣∣∣|ξ |=1 =−

1∫0

L(s,ε)ds+1

2(1− ε).

Let

L2i+1(ε)≡ 0, i= 1,2, . . . ,m− 1, l2m+1(ε) 6= 0

in (8.6.12). Assuming that ξ1 = r cosα, ξ2 = r sinα, let us reduce the system (8.6.12)to the single equation

ε

1− ε+L2m+1(ε)r

2m+O(r2m+2)= 0. (8.6.13)

Note that

L2m+1(0)= λ0 < R2m+1(ξe),ej > ξ−1j |ξ |

−2m, j= 1,2,

for all ξ if R2u= ·· · = R2m(u)= 0.

Remark 8.12. If R2(u) 6= 0, then for all ξ we have

L3(0)= λ0ξ−1j |ξ |

−2 < R3(ξe)+R′2(ξe)0R2(ξe),ej >, j= 1,2.

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From equation (8.6.13) we find two solutions

r1,2 =±2m

√−ε

L2m+1(0)+O(|ε|1/2m),

which are real for εL2m+1(0) < 0. The two solutions∣∣∣∣ ξ1ξ2

∣∣∣∣= (± 2m

√−ε

L2m+1(0)+O(|ε|1/2)

)∣∣∣∣ cosαsinα

∣∣∣∣ (8.6.14)

of the BEq correspond to these solutions, where the parameter α ∈ R correspondingto the group O(2) remains arbitrary. Substituting (8.6.14) into (8.5.5) and the vector(8.5.5) into (8.6.10), we obtain two solutions

[ϕ±

ψ±

]=

−2αq

mu01

q

mcu02

±±

[−T2/T1

1

]js0(µ

(s0)σ0ρ)coss0(2−α)

2m

√−

λ− λ0

L2m+1(0)+ 0(|λ− λ0|

1/(2m)).

For (λ− λ0)L2m+1(0) < 0, to these solutions there correspond two real solutionsf±i ,E

±,B± of the stationary VM system (8.1.1)–(8.1.2) with boundary conditions(8.1.3), (8.1.4) determined by equations (8.4.4).

In conclusion, we note that instead of condition (8.4.11) it sufficies to impose thefollowing condition:

(v) the potentials U, A of the electromagnetic field (E, B) satisfy conditions (8.1.3),and, hence,

N∑i=1

qiAi(liϕ0,kiψ

0)= 0,N∑

i=1

qi(βi,d)Ai(liϕ0,kiψ

0)= 0,

where ϕ0, ψ0—are harmonic functions with the boundary conditions

ϕ0|∂D =−

2αq

mu01(r), ψ0

|∂D =q

mcu02(r).

Moreover, if distribution functions has the form

fi = fi(a(−αv2+ϕi)+ b(vdi+ψi)),

then results are obtained by analogy and in case of inconstant values u01, u02.The procedure presented can be used for constructing the nontrivial solutions of

an integro-differential system (8.4.10). Therefore, analogous results occur, also, in theproblem about the point of bifurcation of VM system with normalized distributionfunctions.

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9 Boltzmann Equation

9.1 Collision Integral

Describing the difference between Vlasov and Boltzmann equations, one shouldremember that Vlasov’s equation describes long-range action, while Boltzmann’sdescribes the short-range action, i.e., how the particle collision processes behave.Vlasov equation describes a shift of distribution function along trajectories of dynamicsystem, while trajectories also depend from the distribution function. The Boltzmannequation takes into account the number of collisions in a cell of the typical size. This isan evolutionary equation for distribution function f (t,x,v) of the particles over veloc-ities v and space coordinates x describing in-flow and out-flow of colliding particles

∂f

∂t+

(v,∂f

∂x

)= I[ f , f ]. (9.1.1)

Assuming the right part of the equation (9.1.1) to be zero, one obtains the equation offree motion. The function I[ f , f ] describes collisions and is called collision integral orcollision operator. The word operator is used to outline the fact that we establish thecorrespondence between functions f and I[ f , f ]. The last represents quadratic operator,which acts as distribution function via variable v only, since x and t are assumed to beparameters [52, 53, 64, 67]:

I[g1,g2](v)=∫ [

g1(v′)g2(w′)− g1(v)g2(w)]×B

(|u|,

(u,n)|u|

)dndw.

Here n ∈ S2—the unit sphere vector, indicating a direction of relative particle veloc-ity after a collision u= v−w—relative velocity; v′ and w′—particles velocities aftercollisions:

v′ =v+w

2+|u|n

2, w′ =

v+w2−|u|n

2. (9.1.2)

In order to represent or describe the process of collisions, the diagram of collisionsis drawn. First, we draw the sphere of radius R= |u|2 with center coordinates u+w

2 . (seeFigure 9.1).

For the given precollision velocities v and w, all possible opposite points of thissphere v′ and w′ are the particle velocities after collisions. We note that relation


http://dx.doi.org/10.1016/B978-0-12-387779-6.00009-0

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v

w ′

w

v ′

Figure 9.1 Collision diagramm.

v+w= v′+w′ as conservation of impulse and v2+w2

= v′2+w′2 as conservationof energy are the basis for deriving (9.1.2). The similar expression will be obtainedbelow for the particles of different masses.

Function B(|u|,µ) is a product relative velocity module |u| and differential scatter-ing cross-section σ(|u|,µ): B(|u|,µ)= |u|σ(|u|,µ).

For power potentials U(r)= 1rα , one has B= |u|1−4/αgα(µ). Potential U(r)= r−4,

α = 4 is independent from |u| function B and is called a Maxwellian molecular poten-tial. Under this assumption the expression for collision integral is simplified andMaxwell was able to derive the equations on macroscopic characteristic variables:density, velocity, and temperature. This successful result was quite unexpected, so hetried to prove that all natural forces act as minus fifth degree of distance. This fact wasnot justified later in experiments: simplest solutions do not always describe the realnatural phenomena behavior. Here n= 5 defines a relation between force and distanceand corresponds to α = 4 for the potential.

9.2 Conservation Laws and H- Theorem

Let us study the evolution of integral characteristics 8ϕ =∫ϕ (v, f (t,v,x))dvdx for

the Boltzmann equation, regarding some function ϕ(v, f ). Using (9.1.1), one obtains

d8ϕdt=

∫ϕ′f∂f

∂tdvdx=−

∫ (v,∂f

∂x

)ϕ′f dvdx+

+

∫ϕ′f I[ f , f ]dvdx= S1+ S2. (9.2.1)

Denote ϕ′f = ψ . Now we can transform the second term

S2 =

∫ψ(v)[ f (w′)f (v′)− f (w)f (v)]B

(|u|,

(u,n)|u|

)dwdvdndx.

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Boltzmann Equation 167

We transform the above expression identically, exchanging v↔ w and obtainingsymmetric expression on S2, adding two equal terms with coefficient 1/2. Next,we exchange (v,w)↔

(v′,w′

). Therefore, the element dwdvdn becomes dw′dv′dn′.

Taking into account that v′ and w′ are linearly dependent on v and w,(v′

w′

)= A

(vw

),

due to (9.1.2), one finds the Jacobian J(A) of transformation A equal to the deter-minant A: J(A)= det A, dw′dv′ = |det A|dwdv. Here A2

= I, hence (detA)2 = 1, and,therefore, dw′dv′ = dwdv.

Thus, we obtain two more terms and coefficient 1/4:

S2 =1

4

∫ (ψ(v)+ψ(w)−ψ(v′)−ψ(w′)

)×[f (w′)f (v′)− f (w)f (v)

]×

×B

(|u|,

(u,n)|u|

)dwdvdndx. (9.2.2)

Example 9.1. Calculate detA.Hint: Writedown matrix A in the form

A=(

E− p pp E− p

),

with p= pu = (u,m)m—one-dimensional projector onto vector m:

v′ = v+ (u,m)m,

w′ = w− (u,m)m,

u= v−w.

This expression is useful as an additional form, representing collisions (9.1.2).Answer: detA=−1.Using formulae (9.2.2), we can introduce some corollaries.

9.2.1 Conservation Laws

If we assume

ψ(v)+ψ(w)= ψ(v′)+ψ(w′),

then S2 ≡ 0. The given equation posesses five-dimensional solution space over con-tinuous functions

ψ(v)= φ05(v)= av2+

3∑i=1

bivi+ c

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(see [52, 64, 67] for details). Here a, bi (i= 1,2,3), c are arbitrary constants. Thesetypes of solutions are called summator invariants or additive invariants of collisions.The solution basis corresponds to the number of particles

∫f dvdx, three components

of velocity vector∫

vi f (v,x)dvdx and kinetic energy∫

v2f (v,x)dvdx.The expression φ05(v) denotes 5 as a summator invariants space dimension, and 0

means conservation.

9.2.2 Boltzmann H- Theorem

Assuming ψ(v)= ln f , we obtain

S2 =1

4

∫ln

(f (v)f (w)

f (v′)f (w′)

)[ f (v′)f (w′)− f (v)f (w)]Bdvdwdndx.

In this integral function, B is nonnegative and strongly positive almost everywhere.Since the first two factors have different signs, one obtains S2 ≤ 0, that corresponds tothe first part of H- theorem [52].

The second part consists of condition, providing equility S2 = 0. Observing non-negativity of integrand function and using inequility B> 0 strongly almost every-where, one obtains an equality condition f (v′)f (w′)= f (v)f (w). Taking the logarithmof this equality, we get ln f (v) as a summator invariant: ln f (v)= φ05(v). Hence,f = f0 = Ae−β(v−v0)

2.

Using the physical sense, we treat A> 0, because a distribution function is a non-negative one; we also take β > 0, since negative β means an increasing of distribu-tion function in infinity and gives divergent expressions for local velocity, densityof particles and temperature; β = 1

2kT is proportional to inverse temperature, k—isa Boltzmann constant, v0—mean velocity. Parameter A is convenient to write in the

form A= ρ[2πkT]−32 , treating quantity ρ as density. All these parameter quantities

ρ, T and v0 could be coordinate x and time t dependent. Such distribution is calledlocal-Maxwellian.

H- theorem means that the functional H =∫

f ln f dvdx decreases in time due tocollision: S2 ≤ 0. And S2 = 0 if the distribution function is local Maxwellian.

9.2.3 Beams

Generally speaking, the local-Maxwellian distribution is not an exact solution ofBoltzmann equation: free motion “spoils” it. Boltzmann carefully investigated suchcases of exact solutions [52, 67]. We give one special example for T = 0, producingbeams. Please note that, while temperature T tends to zero and density ρ(x, t) keptunchanged, the local-Maxwellian distribution transforms into f (t,x,v)= ρ(x, t)δ(v−v0(x, t)).

We are already studied such distributions (see Chapter 2, Section 2.5, called hydro-dynamic substitution). They describe the beams of particles. Now we see that collisionintegral vanishes for such distributions and also gives exact solutions of Boltzmann

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equation if ρ and v0 satisfy the system of equations of free motion (see Sections 2.5.1,2.5.2 with K = 0). If the number of beams is greater than 1 (see N-layer hydrodynam-ics section of Chapter 2), then collision integral do not vanish.

9.2.4 Validity Conditions for Conservation Laws and H- Theorem

Let us consider now the first term S1 in equality (9.2.1). Transforming it to the integralfrom divergence operator

S1 =−

∫∂

∂xi(viϕ)dvdx,

we found it trivial in many cases. Let us consider three of them:

Infinite space

Here we transform the above expression integrating every term over its derivative. Weobtain

S1 =−

∫v1[φ(x1 =+∞,x2,x3)−φ(x1 =−∞,x2,x3)]dx2dx3dv+

+ two similar terms.

Hence, if it is seen that f (t,v,x) tends to the same function g(t,v), independent ofvariable x, then S1 = 0. Vise versa, if f tends to different functions, then there appearsto be the flows of the corresponding values. Thus, both inequality of H- theorem andconservation laws could be violated. A well known typical example is the shock wave,when the distribution function tends to different Maxwellian distribution at x→±∞.

Periodical problem

Let f (t,v,x+ (n,T))= f (t,v,x) for some linear independent vector set T = (T1,T2,

T3) and for all vectors n= (n1,n2,n3) with integer components. Then, integratingover periods, one obtains conservation laws and H- theorem on the basis of equalityS1 = 0. Therefore, the conservation laws and H- theorem are fulfilled in periodicalproblem.

Boundary-value problem

If x ∈ D, where D is domain with a smooth boundary ∂D, then, integrating over thatdomain and using the Ostrogradsky formulas, we find

S1 =−

∫D

∂

∂xi(vi,ϕ)dvdx=−

∫∂D

(v,n)ϕdvdS.

Hence, the equality to zero of the flows through the boundary of correspondingvalues is necessary for global conservation laws.

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The most popular boundary conditions [67] are Maxwell conditions:

1. specular reflection: f (x,v, t)= f (x,v− 2n(n,v), t), x ∈ ∂D, (v,n) > 0, where n—vectorof normal to surface in the point x;

2. diffuse reflection: when we obtained locally-Maxwellian distribution under reflection.

One may find details in [64, 67].

9.3 Boltzmann Equation for Mixtures

Consider mixture of two gases with masses of molecules M and m. Let F(t,p,x)and f (t,q,x) distribution functions of molecules of heavy and light kinds, respec-tively, defined by impulses ( p ∈ Rd, q ∈ Rd, d = 1,2,3), space x ∈ Rd in momentt. Because of impulse conservation for particle collisions, it is convenient to con-sider impulses instead velocities. The Boltzmann equation is reduced to the followingsystem:

∂f

∂t+

(q

m,∂f

∂x

)= I[F, f ]+ I[ f , f ],

∂F

∂t+

(p

M,∂F

∂x

)= I[ f ,F]+ I[F,F],

(9.3.1)

where I[F, f ]—has the form

I[F, f ]=∫

[F(p′)f (q′)−F(p)f (q)] |u|σ

(|u| ,

(un)

|u|

)dpdn.

Here u= (p/M)− (q/m)—relative velocity, p′, q′—impulses after collisions:

p′ =p+ q

2(1+ δ)+

1

2|u|µn,

q′ =p+ q

2(1− δ)−

1

2|u|µn,

(9.3.2)

where δ = (M−m)/(M+m), µ= (2Mm)/(M+m)—mean harmonic of masses(2/µ= 1/M+ 1/m); n—vector of unit sphere: |n| = 1, n ∈ Rd.

If we define p′ and q′ in this way, they satisfy conservation law for impulsep′+ q′ = p+ q and energy

(p2/2M)+ (q2/2m)= ((p′)2/2M)+ ((q′)2/2m).

Let us show that parametrization (9.3.2) is obtained from conservation laws. Puttingp+ q= Q and expressing impulses of heavy particles p′ = Q− q′ and p= Q− q fromconservation law of impulse, we substitute them into conservation law of energy. After

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q

AO

O′

p

p′

q ′

Figure 9.2 Collision diagramm.

transformations, we have(q′−

Q

αM

)2

=

(q−

Q

αM

)2

,

where α = (M+m)/(mM)= 2/µ.For q′, this is a circle equation with a center in

Q/(αM)= ((p+ q)/2)(1− δ)

and radius

R2= (q−Q/(αM))2 = |µ|2|u|2/4.

We obtain a similar expression for heavy particles. The collision scheme shown inFigure 9.2 explains that impulses of the light and heavy particles define the spheres ofequal radius (|µ| |u|)/2 with their centers to be symmetrical with respect to (p+ q)/2.

Here p and q—impulses of particles before collision, p′, q′—after; A—mean ofinterval p q and p′ q′; O—center of the sphere for impulses p′ of heavy particles withcoordinate ((p+ q)/2)(1+ δ); O′—center of sphere of impulses q′ of light particleswith coordinate ((p+ q)/2)(1− δ).

Also the invariants and H- theorem with derivation of Maxwellian distribution forboth components are considered. For these, the temperatures are found the same,and density—vary. Passing to the limit T→ 0, we obtain f (q,x, t)= ρ(x, t)δ(q−mV(x, t)), F(q,x, t)= η(x, t)δ(q−MV(x, t)). Therefore, if functions ρ,η,V satisfy thesystem of equations:

∂ρ

∂t+

∂

∂xi(Viρ)= 0,

∂η

∂t+

∂

∂xi(Viη)= 0,

∂Vi

∂t+

∂

∂xk(ViVk)= 0,

then they are exact solutions of system (9.3.1) and the system of free movement beforeoverlapping.

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9.3.1 Chapman-Enskog Method and Hydrodynamics in TwiceDivergent Form

To be able to use hydrodynamic methods, we consider a Boltzmann equation with alarge parameter multiplier for collision integral:

∂f

∂t+

(v,∂f

∂x

)=

1

εJ(f , f ).

We assume f = f0+ εf1+ . . . Hence, one obtains J(f0, f0)= 0 and f0 = e(ϕ,α) islocally Maxwellian distribution, ϕ is vector of five invariants, α(x, t) are parameters,through which the density, mean velocity, and energy are expressed for (explicit for-mulas see Section 9.5). Thus, we put

ϕ0 = 1, ϕ1 = v1, ϕ2 = v2, ϕ3 = v3, ϕ4 = v2.

9.3.2 Hilbert Method

As the first approximation, we find

∂f0∂t+

(v,∂f0∂x

)= L( f0, f1), where L(f0, f1)= J(f0, f1)+ J(f1, f0).

To solve an obtained equation with respect to f1, an orthogonality condition for allinvariants of L is required. It gives us equations for parameters α leading to Eulerequations. We may write them down in the following twice divergent form [116, 118]:

∂Lαµ∂t+ div EMαµ = 0, µ= 0, . . . ,4, (9.3.3)

where L(α)=∫

exp(α4v2+

3∑i=1αivi+α0)dv,

EM(α)=∫Evexp(α4v2

+

3∑i=1

αivi+α0)dv.

Here α4 < 0, Lαµ—partial derivative in αµ. We see that equations are written viafunctionals L(α) and EM(α). This is twice divergent (Gogunov’s) form. Here and laterµ= 0,1, . . . ,4; i= 1,2,3. This form is very important, due to hyperbolicity of equa-tions for the first approximation. Also one can consider it a basis for nonequilibriumGibbs method (see details in sections 9.4, 9.5).

9.3.3 Chapman-Enskog Method

∂

∂t=

∂

∂t0+ ε

∂

∂t1,

i.e., two time variables are introduced—for fast and slow times.

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We will not discuss these methods, because all corresponding material exists inmany textbooks.

9.4 Quantum Kinetic Equations (Uehling-UhlenbeckEquations)

We may write quantum kinetic equations in the form similar to Boltzmann’s (1.1.1):

∂f

∂t+

(v,∂f

∂x

)=

∫BT( f )[h(v′)h(w′)− h(v)h(w)]dw,

where as before B= |u|σ(|u| , (u,n)/ |u|);h(v)= f (v)/(1+ θ f (v)), θ = 1 for bozons,θ =−1 for fermions, θ = 0 for Boltzmann equation (1.1.1);

T( f )= (1+ θ f (v))(1+ θ f (w))(1+ θ f (w′))(1+ θ f (v′))=

=f (v)f (w)f (w′)f (v′)

h(v)h(w)h(w′)h(v′);

H- function is given by formulas

H( f )=∫

[ f ln f − (1+ θ f ) ln(1+ θ f )],

with steady-state Boze and Fermi distributions defined from the condition

f0(v)/(1+ θ f0(v))= e(ϕ,α), i.e., f0(v)= e(ϕ,α)/(1− θe(ϕ,α)).

One can also write the corresponding hydrodynamics equations in twice divergentform. Also, more generally, when the stationary state of collision integral is an arbi-trary function g of (ϕ,α), i.e.,

f0 = g(∑

ϕµ(v)αµ), (9.4.1)

we have the corresponding Euler equations in the form (9.3.3) for

L(α)=∫

h(∑

ϕµ(v)αµ)

dv, (9.4.2)

where h(λ) is antiderivative of g: h′(λ)= g(λ).For Uehling-Ulenbek equations:

L(α)=−1

θ

∫ln(

1− θe∑ϕi(v)αi

)dv.

We see the twice divergent form of notation it naturally follows from kinetic equa-tion both in classical and quantum case.

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9.5 Peculiarity of Hydrodynamic Equations, Obtainedfrom Kinetic Equations

In the case of equilibrium functions of general form (9.4.1), we have equations of aEuler type:

∂

∂t

∫αµ f0dv+

∂

∂xi

∫viαµf0dv= 0. (9.5.1)

We run summation by i= 1,2,3. One can rewrite these equations using two func-tions, instead of four in (9.3.3):

∂Lαµ∂t+∂Rαµαi

∂xi= 0, µ= 0, . . . ,4. (9.5.2)

Here Rαµαi —the second partial derivatives in αµ and αi, L(α) is defined by formu-las (9.4.2), and

R(α)=∫

q(∑

ϕµ(v)αµ)

dv,

where q is antiderivative of function h(x) from (9.4.2): q′ = h. Such form of equa-tions appears from ϕ1 = v1, ϕ2 = v2, ϕ3 = v3, i.e., three components of velocity areinvariants for any of the considered collision integrals.

In the case of Boltzmann equation, R= L and equations (9.5.2) are written in termsof unique function L. Thus, variables α are expressed via ordinary macroscopic values(ρ,u,T) in explicit form. It is obtained by means of comparison of two expressionsfor Maxwellian distribution:

f0(v)=ρ

(2πkT)3/2exp

[−(v− u)2

2kT

]= exp

(α0+

∑αivi+α4v2

)=

= exp

(α0−

α2

4α4

)exp

[α4

(v+

α

2α4

)2].

Here α = (α1,α2,α3), α2= α2

1 +α22 +α

23 .

Thus, one obtains

α4 =−1

2kT, αi =

ui

kT, α0 = lnρ−

3

2ln(2πkT)−

u2

2kT.

These variables are typical in kinetic theory, and we can use them in gas dynam-ics as well. They were applied for calculation of specific problems of gas dynam-ics by M. A. Rydalevskaya [249]. In addition, the nonsingularity of that change wasinvestigated.

Such variables were used in paper [9] for reconstruction of formulas for entropyover minimizing distribution of the form (9.4.1). If we seek stationary state fromcondition of minimum of H- function (or maximum of entropy) under conservationcondition for five invariants, then these variables become Lagrange multipliers.

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9.6 Linear Boltzmann Equation and Markovian Processes

The following equation

∂f

∂t= J(F, f )=

∫ [F(w′)f (v′)− f (v)F(w)

]× (9.6.1)

×B

(|u| ,

(u,n)

|u|

)dwdn

is called a linear Boltzmann equation.It describes the situation when the component f has comparatively small density

and one can neglect collisions of particles of that component in comparison with col-lisions with particles of component F: the particles f scatter on particles F. Here F(v)is a given function of variable v. If we consider space-nonuniform situation, then wehave equation

∂f

∂t+

(v,∂f

∂x

)= J(F, f ).

Such equations are common in transport theory of neutrons in nuclear reactors andalso in the problems of radiation described transport [188, 302].

In contrast to Boltzmann equation, (9.6.1) has only one linear invariant for positivefunction F(w)—a number of particles, namely

d

dt

∫ϕ(v)f (v)dv=

∫[ϕ(v′)−ϕ(v)]F(w)f (v)Bdwdvdn.

This expression is equal to zero when ϕ(v) is constant. In order to explain the behaviorof the solution of equation (9.6.1), we rewrite it in the form

∂f

∂t=

∫K(w,v)f (w)dw− a(v)f (v, t)= L( f ). (9.6.2)

Such presentation is accepted in the theory of reactors and the problems of radiationtransport. Kernel K(w,v) and coefficient a(v) can be explicitely written in terms offunctions F and B, but all we need is an equality a(v)=

∫K(v,w)dw, which rep-

resents conservation law for numbers of the particles. One can also rewrite equa-tion (9.6.2) in the form typical for Markovian processes or for chemical kinetics[103, 217]:

∂f

∂t=

∫[K(w,v)f (w)−K(v,w)f (v, t)]dw. (9.6.3)

Here K(w,v) represents the probability of transition from the state w in v(Markovian process jump).

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If ξ(v) is a positive stationary state of equation (9.6.2) (i.e., L(ξ)= 0), then one canprove the following form of H- theorem (see, for example [217]).

Consider function H( f )=∫

f ln[ f (v)/ξ(v)]dv. Due to (9.6.3), derivative(dH/dt)≤ 0 for any positive nuclear K(w,v). Theorem is proven, applying inequalityu lnu− u+ 1≥ 0, where equality is reached for u= 1. Writing expression for dH

dt andexchanging v and w in second term, one obtains

dH

dt=

∫K(w,v)f (w) ln

f (v)ξ(w)

ξ(v)f (w)dvdw=

=−

∫K(w,v)f (w)

1

u[u lnu− u+ 1]dvdw≤ 0. (9.6.4)

Here u= (f (w)ξ(v))/(ξ(w)f (v)) and

∫K(w,v)f (w)

1

u[1− u]dwdv= 0,

that follows from stationary state of ξ(v).Starting from inequality (9.6.4), it follows that solution of equation (9.6.3) tends to

stationary state, which is proportional to ξ(v) with the same integral for the numberof particles. From uniqueness of stationary solution follows the uniqueness for thenumber of particles as a linear integral for positive K(w,v).

Equality in (9.6.4) is reached for u= 1, i.e., f (w)ξ(v) =

f (v)ξ(v) for all points (w,v), where

K(w,v) > 0. The question on behavior of the solution for equation (9.6.3), hence,is reduced to existence of positive stationary solution ξ(v). Sometimes ξ(v) can bewritten explicitly. Thus, if Maxwellian distribution F0 is taken as F in (9.6.1), thenξ(v) is proportional to F0.

Sometimes, digitization of (9.6.3)

∂fm∂t=

∑j

(Kjm fj−Kmj fm

)=

∑j

Ajm fj, m= 1, . . . ,n, (9.6.5)

is said to be basic (or master) Pauli equation. It is studied in detail in the theory ofMarkovian processes [103].

Here matrix Ajm is obtained from matrix Kjm, representing transition probabilityfrom the state j into the state m. Let Kjm be positive for j 6= m and Kjj = 0. Then Ajm

is obtained with positive off-diagonal elements, negative diagonal, and a sum of theelements in every column equals to zero:

∑m

Ajm = 0. Such form of matrix A automat-

ically preserves nonnegativity of fm(t) and keeps the conservation law ddt

(∑fi)= 0

for the total number of particles fulfilled.An opposite is also correct. Any linear system with matrix complying these

two proporties (off-diagonal positiveness and equality to zero of a sum of the ele-ments of every column) allows representation via some matrix Kmj of transition

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probabilities:

Amj = Kmj, m 6= j, Ajj =−∑

m

Kjm.

For system (9.6.5), H- theorem can be proved with the following H- function

H =∑

i

fi lnfiξi,

where ξ—stationary solution. Such solution always exists, because determinant ofmatrix A is null because of conservation law (a sum of all rows equals to zero).

It is important to find stationary solutions ξ of equation (9.6.5) with strongly posi-tive components ζi to comply with H- theorem.

Here we show when it is possible to obtain the stationary solutions in explicitform.

1. Symmetry of matrix Kjm; more general, an equality to zero of the sum of the elements inevery line of matrix Ajm give stationary vector ξ with equal components.

2. The case of detail equilibrium, when there exist such numbers rj that Kmjrm = Kjmrj. Thenstationary vector is proportional to vector r.

Exercise 9.1. Show that all matrices 2× 2 satisfy condition 2), but matrices 3× 3—only for K12K23K31 = K13K32K21. The dimension of matrices Kn×n is n(n− 1), andmatrices K with condition 2)—n(n− 1)/2+ n− 1.

Going back to kinetic equations, many satisfy both properties: positivity and con-servation law for the number of particles, for example, all shift equations from 3.2(Liouville equation: see part 9.7). It is evident that it does not guarantee a convergenceto the stationary solutions. We could try to realize the situation making a discretizationof this Liouville equation. Then we get our system (9.6.5) with sparse matrix Amj.The behavior of this discretization and of original Liouville equations are different.Nevertheless, the discretization idea originated in Markov’s processes theory is ratherproductive. In this way we keep the positive properties of solutions unchanged, and anumber of particles is unchanged. Next section will be devoted to dissipative proper-ties of Liouville equation and clarify such contradictions.

9.7 Time Averages and Boltzmann Extremals

The maximum entropy principle under linear conservation laws gives Boltzmannextremals [51]. The stochastic ergodic theorem [238] asserts the existence of timeaverages or Cesaro means. We prove the coincidence of time averages with Boltz-mann extremals.

9.7.1 Boltzmann Extremals

In [51], Boltzmann proves his H- theorem, which asserted the convergence of solu-tions of Boltzmann-type equations to a Maxwellian distribution. In [51] Chapter 2, the

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maximum entropy principle was stated. Boltzmann considers the system of equations:

du1

dt= B(u2

2− u1u3)

√2

du2

dt= B(u1u3− u2

2) (9.7.1)

√2

du3

dt= B(u2

2− u1u3).

Boltzmann shoed that the functional

E(u)= u1lnu1+√

2u2lnu2+√

3u3lnu3 (9.7.2)

decreases by virtue of system (9.7.1), that is,

dE

dt= B(u2

2− u1u3)lnu1u3

u22

≤ 0. (9.7.3)

This leads one to conclude that the solutions of the system converge to their station-ary. In searching for a stationary with respect to the initial condition, Boltzmann usesinvariant linear functionals. In the case of (9.7.1), these are

A(u) = u1+√

2u2+√

3u3 = a, and

B(u) = u1+ 2√

2u2+ 3√

3u3 = b.(9.7.4)

To determine the limit to which any solution of (9.7.1) converges, we must minimizefunctional (9.7.2) subject to (9.7.4), where the constants a and b are determined fromthe initial conditions. The stationaries obtained from this variational principle are pre-cisely the Boltzmann extremals.

9.7.2 Boltzmann Extremals and the Liouville Equation

Consider the system of n ordinary differential equations

dx

dt= v(x). (9.7.5)

Here x= (x1,x2, . . . ,xn) and vi(x) are continuously differentiable functions. Considerthe continuity (Liouville) equation for this system:

∂f

∂t+5fv(x)= 0. (9.7.6)

Suppose that system (9.7.5) has a solution on the entire time domain (i.e., a globalsolution) and is divergence-free (i.e., 5v(x)= 0). Then, we can write the solution of(9.7.6) in the form f (t,x)= f (0,g−t(x)), where g−t(x) is the shift of the point x in timet by virtue of system (9.7.5). We define the time averages, or Cesaro means, of the

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solution (9.7.6) by

fT(x)=1

T

T∫0

f (t,x)dt. (9.7.7)

Neumann’s stochastic ergodic theorem asserts that the limit f C of the function fT as Ttends to infinity exists in L2(Rn) for any initial data from this space.

We define entropy as a functional on the set of positive functions h(x) from LL2(Rn) by S(h)=−

∫hlnh(x)dx. Such functionals are invariant for (9.7.6) in the

divergence-free case. However, in [231], Poincare discussed the entropy growthfor the limit function in the particular example of a collisionless gas. Kozlov andTreshchev generalized this result (see [159]); namely, they proved that the entropyof the time average is not less than that of the initial distribution for (9.7.6). Inthis section, we show that the solution of (9.7.6) converges “where it must:” thetime averages are determined by the conditional maximum entropy principle (theBoltzmann principle). The first attempts to apply the Boltzmann principle to thissituation were made in [297].

We define linear conservation laws for the continuity equation (9.7.6) as the linearfunctionals Iq(h)=

∫q(x)h(x)dx= (q,h), which are invariant by virtue of (9.7.6), i.e.

(q, f (x, t)) does not depend on time on the solutions f (t,x) of (9.7.6). Let I denote theset of such functions q from L2 (integrals).

Consider the Cauchy problem for (9.7.6) with positive initial conditions f (0) fromL2(Rn). We define the Boltzmann extremal f B

= f B(f (0)) as a function maximiz-ing the entropy subject to the given conservation laws: S(f B)=maxS(h) on the setL(I, f (0))= {h : (q,h− f (0))= 0 ∀q ∈ I}.

Theorem 9.1. If S(h) (entropy) tends to minus infinity when norm of h tends to infinityon the set L(I, f (0)), then

(i) the Boltzmann extremal exists and is unique;(ii) the Cesaro mean coincides with the Boltzmann extremal, i.e.,

f Ch= f B. (9.7.8)

This theorem follows from a more abstract result obtained further.

9.7.3 Boltzmann Extremals and the Ergodic Theorem

Let U be a linear operator on a Hilbert space X with norm not exceeding 1. Then,the following theorem is valid, which is known as the stochastic ergodic theorem (ofRiesz; see [238]). For each z from X, the time averages

Pnm(z)=1

n−m

n∑k=m+1

Ukz

strongly converge to some element PC= PC(z) as n−m tends to infinity. This element

is invariant with respect to U, i.e., U(PC)= PC. This is what is known as the stochastic

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ergodic theorem. It determines the time averages, or Cesaro means, PC= PC(z)=

limn−m→∞

Pnm(z).

We define a Boltzmann extremal as an element PB for which the entropy attains aconditional maximum. More precisely, consider the set I ⊂ X consisting of linear con-servation laws u ∈ I such that (Ux,u)= (x,u) for all x ∈ X. Let S(x) be a strictly con-cave (convex upward) functional not decreasing under the action of U, i.e., such thatS(Ux)≥ S(x) (this is an analogue of the entropy), and let Xz be the set of x ∈ X with thesame constants of linear conservation laws as z, i.e., Xz = {x ∈ X : (x− z,u)= 0 ∀u ∈I}. Consider the conditional extremal problem of finding a point at which supS(x)is attained subject to the constraint x ∈ Xz. We denote this conditional extremum byPB(z) and call it a Boltzmann extremal.

Theorem 9.2. If S(x) tends to minus infinity when norm of x tends to infinity on

(i) the extremal problem stated above has a unique solution in Xz, and(ii) the Cesaromeans coincide with the Boltzmann extremals, i.e.,

PC(z)= PB(z). (9.7.9)

Proof. Here we outline the scheme of the proof:Any such functional S (tending to minus infinity when norm of x tends to infinity

and strictly concave) has a unique maximum on any linear closed subspace of aBanach space. Since Xz is closed, this proves the first assertion. Since Pnm(z) ∈ Xz,it follows that PC(z) ∈ Xz. Therefore, S(PB(z))≥ S(PC(z)). Thanks to the uniquenessof the extremal, it only remains to show that this inequality is an equality. Follow-ing [238], consider two subspaces in X. One subspace, Y , consists of the elementsx−Ux and their limits. The other subspace, Z, consists of the fixed elements: x2 ∈ Ziff U(x2)= x2. It was proven in [238] that X is the direct sum of these spaces. There-fore, all points of Xz have the same time average, and, hence, PC(z)= PC(PB(z)). Theconvexity of S implies that, for any z, we have

S(Pnm(z))= S

1

n−m

n∑k=m+1

Ukz

≥ 1

n−m

n∑k=m+1

S(

Ukz

)≥

≥1

n−m

n∑k=m+1

S(z)= S(z),

therefore, S(PC)(z))≥ S(z). Applying this relation to PB(z), instead of z, we obtainS(PC(z))= S(PC(PB(z)))≥ S(PB(z)). Hence, the reverse inequality is proven and,thereby, the coincidence of the time averages with the Boltzmann extremals.

Theorem 9.1 is obtained from Theorem 9.2 by considering shifts in unit time andrestricting the problem to the invariant cone of positive functions. Note that the right-hand side of (9.7.9) depends on the functional S, whereas the left-hand side does not.At the same time, the left-hand side depends on the operator U, whereas the right-handside depends on it only via the conservation laws and the functional S being increasing.

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It follows that Theorem 9.1 remains valid when concave functionals, instead of theentropy, are taken.

Let us show that tending to infinity of the functional on the level sets of the linearconservation laws is essential.

Example 9.2. Consider the simplest Liouville equation, which corresponds todx

dt= 1,

namely,

∂f

∂t+∂f

∂x= 0.

Consider its solution f (t,x)= f (0,x− t) in L2(R). A step of high a and width K hasentropy—Kalna, but its Cesaro mean vanishes. There are no invariant linear func-tionals (in L2); therefore, the Boltzmann extremal is infinity (to show this, it sufficesto take positive a less than 1 and let K tend to infinity for this step). Thus, Theorems 9.1and 9.2 are false in this case, because the entropy is not bounded above in L2 (even onsteps). Note that, for numbers a smaller than 1, the entropy decreases. For a periodicproblem (an equation on a circle), one conservation law (the number of particles)arises. On the corresponding set, the entropy is bounded above, and the Cesaro meandoes coincide with the Boltzmann extremal.

Example 9.3. For the same equation and the functional “minus norm” the conditionsof the theorem holds even on the line, and the Boltzmann equal to Cesaro mean.

Example 9.4. (The finite-dimensional case.) The spectrum of the operator U inTheorem 9.2 must be located on the unit disk. The conservation laws correspond tothe zero unit point of the spectrum.

Example 9.5. (Hamiltonian systems with compact energy levels.) In this case, theentropy functional is bounded above on functions with fixed energy, and the condi-tions of Theorem 9.1 hold; therefore, the time average coincides with the Boltzmannextremal.

Linear conservation laws are related to partitions into ergodic components [301].It was proven in [159] that, if a solution to (9.7.6) converges, then its limit coincideswith the time average. The results obtained above imply that the limit is determinedby the conservation laws.

Theorems 9.1 and 9.2 are also subject to generalization. They are fulfilled in finite-dimensional case for Markovian processes and their nonlinear generalizations fromchemical kinetics (type (9.7.1) equations). A continuous time case was studied in [297](Boltzmann extremal convergence). Cesaro means are not necessary, since a time limitat infinity exists. Considering discrete time, Markovian processes sometimes do notconverge, but Cesaro means should coincide with Boltzmann extremals. This detail isof great importance for computer modeling, since it always uses a discrete time scale.

In the case of discrete time the convergence to the equilibrium state derived byBoltzmann from H- theorem (one of the fundamental principles) is also violated.

Another application can be seen in [301]—ergodic hypotesis. Even the classi-cal example, hard balls in the box, states that a time limit (infinite time) relates a

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convergence of distribution function with a function depending from energy. In reality,we are studying two theoretical issues: establishing the convergence and calculat-ing the corresponding limit. The first issue is easily solved by a Poincare theo-rem [159, 238] and corresponding integrals. Hence, we can focus our attention ontothe Liouville equation (9.7.6) related to the system (9.7.5) with dx/dt= v · dv/dt= 0.Vectors x,v ∈ R3N where N is a number of the balls. Combining with a boundaryreflection condition

0< xi < l, (xi− xj)2 > d2, xi ∈ R3,vi ∈ R3, i= 1 · · ·N,

where d is a ball diameter, l is the box length, we have to prove that in L2 such prob-lems have solutions of the form f (x,v)= g(v2). In other words, they depend on energy.This reduction to the integrals in ergodic problem “statement” can also be applied toa variety of problems, if we use Theorems 9.1 and 9.2.

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10 Discrete Models of BoltzmannEquation

10.1 General Discrete Models of Boltzmann Equation

Let fi(t,x) distribution function of particles in the space x ∈ <d at the moment t, mov-ing with velocity vi ∈ <

d. Let us also assume that their interactions define a speed ofthe increase of distribution function as Fi( f1, . . . , fn). Then we obtain the system ofdifferential equations in partial derivatives

∂fi∂t+

(vi,∂fi∂x

)= Fi( f1, . . . , fn), i= 1, . . . ,n. (10.1.1)

This system is called the discrete model of Boltzmann equation, if Fi representscollision integral:

Fi( f1, . . . , fn)=∑k,l,j

σijkl( fk fl− fi fj). (10.1.2)

10.2 Calerman, Godunov-Sultangazin, and Broadwell Models

10.2.1 Calerman Model

Let the space dimension d = 1, the number of particles n= 2 and the interacting par-ticles of the same kind generate the particles of the opposite kind, i.e., one interaction(2,2)↔ (1,1) is directed.

∂f1∂t+ v1

∂f1∂x= f 2

2 − f 21

∂f2∂t+ v2

∂f2∂x= f 2

1 − f 22 .

There is no conservation, neither impulse nor energy, for that model.The Calerman model is a good example of the essence of a Boltzmann equa-

tion. It describes a mixture of “competing” processes: relaxation and free motion.


http://dx.doi.org/10.1016/B978-0-12-387779-6.00010-7

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Relaxation tries to equate f1 and f2 to make them “Maxwellian.” Free motion drivestwo distribution functions in different directions. The Chempen-Enskog approxima-tion corresponds to a very rapid relaxation, when f1 is almost equal f2 and motion isdescribed via ρ = f1+ f2. The opposite situation—slow relaxation (a small quantityof collisions)—is called Knudsen gas and corresponds to a large Knudsen number Kn(ratio of a mean free path l to specific length D of a system Kn = l/d is called Knudsennumber).

10.2.2 Broadwell Model

Let d = 2,n= 4, directions of velocities be given by a cross (Figure 10.1).

v3

v1

v4

v2

Figure 10.1 Velocities directions, Broadwell model.

The first and the second kind of particles collide, giving birth to the third and fourthkinds and vice versa: therefore, conservation laws of impulse and energy are satisfied.We have the system of equations:

∂f1∂t+∂f1∂x= f3 f4− f2 f1

∂f2∂t−∂f2∂x= f3 f4− f2 f1

∂f3∂t+∂f3∂x= f1 f2− f3 f4

∂f4∂t−∂f4∂x= f1 f2− f3 f4.

10.2.3 Godunov-Sultangazin Model

Let d = 1,n= 3. The directions of velocities are given by the following scheme,shown on Figure 10.2:

v3 v2 v1

Figure 10.2 Velocities directions, Godunov-Sultangazin model.

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Discrete Models of Boltzmann Equation 185

Under interaction of the first and third kinds, we obtain the second kind. Then wehave the system:

∂f1∂t+∂f1∂x= f 2

2 − f1 f3

∂f2∂t= 2

(f1f3− f 2

2

)∂f3∂t−∂f3∂x= f 2

2 − f1 f3.

Therefore, impulse is kept, but energy is not conserved.The properties of described models will be studied later.

10.3 H- Theorem and Conservation Laws

10.3.1 Carleman Model

Let H =∫

[ f1 ln f1+ f2 ln f2]dx. More general, let φ( f ) be some given function andfunctional be defined by

Hφ =∫

[φ( f1)+φ( f2)]dx.

Calculate the velocity of increase:

∂Hφ∂t=

2∑i=1

∫φ′(fi)

∂fi∂t

dx=−∫ 2∑

i=1

φ′(fi)

(vi∂fi∂x

)dx+

+

∫ [φ′( f1)−φ

′(f2)][

f 22 − f 2

1

]dx= H1+H2.

The first term is transformed into integral from total derivative:

H1 =−∑∫

∂

∂x(viφ( fi))dx,

which vanishes in the same three cases that were stated for the Boltzmann equation.The second integral is equal to zero if φ′ = Const. Thus, this is conservation law forthe particles. The second integral is less than or equal to zero, if φ′′ > 0, i.e., function φ

is convex. Function f ln f is convex, because (f ln f )′′ =1

f> 0, therefore, H- theorem

is satisfied. Here we also possess nontrivial analogs of H- function, since any convexfunction is suitable.

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10.3.2 Broadwell Model

For H =∑

i

∫fi ln fidx, we obtain

dH

dt=−

∑i

∫∂

∂x(vi fi ln fi)dx+

∫( f3 f4− f2 f1) ln

(f1 f2f3 f4

)dx.

Here the second term is less than or equal to zero, but it is not possible to substituteanother convex function, instead of f ln f , to preserve this inequality.

For Godunov-Sultangazin model H- theorem is proven in the same manner.Since the right side of the general discrete model is obtained by summation from

these three basic components, hence, H- theorem for (10.1.1), (10.1.2) follows imme-diately from that fact. Moreover, one may consider a Boltzmann equation as the“continuous sum” of Broadwell models.

10.4 The Class of Decreasing Functionals for DiscreteModels: Uniqueness Theorem of the BoltzmannH- Function

First, we have to define what kind of functionals

HG =

∫G( f1, . . . , fn)dx

decrease for discrete models of Boltzmann equation. The velocity of increase of suchfunctional reads

dHG

dt=

∑i

∫∂G

∂fi

(−vi

∂fi∂x

)dx+

∑i

∫∂G

∂fiFidx= H1+H2.

We want this expression to be nonnegative for all functions {fi(x)} = f (x), i.e., for allperiodical functions or functions equal to zero outside the initial ball

H1+H2 ≤ 0. (10.4.1)

Consider the function f α(x)= f (αx). Making the change αx= y, we see that thefirst and second terms are transformed in different manner:

H1(

f α)=

α

|α|dH1(f )

H2(

f α)=

1

|α|dH2(f ).

Hence, H1(f ) must be equal to zero, otherwise we can choose magnitude and signof α so that inequality (10.4.1) will be violated. Therefore, H1(f )= 0 for all f (x) > 0,f ∈ C2

(Rd), f is finite.

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The study of the condition H1(f )= 0 means finding the functionals conserved forfree motion.

This condition∑i

∫∂G

∂fi

(vi,∂fi∂x

)dx= 0

in one-dimensional case is a condition of the equality to zero of work in field of forcesF(f )i = vi

∂G∂fi

over any closed contour (periodicity of f or equality of zero for large inmodules values x denotes closure of contour).

The last is satisfied, if the field of forces is potential:

∂Fi

∂fj=∂Fj

∂fi, i.e.

(vi− vj

) ∂2G

∂fi∂fj= 0,

or the second partial derivatives G are equal to zero for i 6= j. Hence, ∂G∂fi= ψi (fi) is

a function of one variable. Therefore, G(f )=n∑

j=1φj(fj)

is a sum of functions of one

variable.Investigation of collision integral for different models shows:

1. for Calerman model φ1 = φ2 = φ and any convex function is suitable.2. for Broadwell model and Godunov-Sultangazin models theorem on uniqueness of H- func-

tion is valid.3. a similar reasoning also applies for Boltzmann equation. The study of collision integral

shows that here we also have uniqueness of Boltzmann H- function [285].

10.5 Relaxation Problem

The problem on space-uniform tending of the distribution function to Maxwelliandistribution is called relaxation problem. For discrete models of Boltzmann equation,we have the system of ordinary differential equations:

dfidt=

∑klj

σijkl

(fk fl− fi fj

).

For H =∑

i fi ln fi we have inequality (H- theorem):

dH

dt=

1

4

∑k,l,i,j

σijkl

(fk fl− fi fj

)ln

(fi fjfk fl

)≤ 0.

From H- theorem follows that any nonnegative initial distribution tends to equilib-rium cone given defined by equalities fi fj = fk fl.

Taking the Calerman model as an example, we get a conservation law of theform f1+ f2 = ρ. Hence, a solution becomes ( f1− f2)= ( f1− f2)(0)e−ρt. Therefore,it tends to the “Maxwellian distribution” cone for f1 = f2 (see Figure 10.3).

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f2

f1

f1= f2

Figure 10.3 Relaxation for Carleman model.

Exercise 10.1. Solve space-uniform Broadwell and Godunov-Sultangazin models andconsider relaxation for them.

It can be proved for Boltzmann equation that any space-uniform distribution tendsto the Maxwellian. The first strong results of the kind were obtained by T. Caler-man [64]. The modern state of the question can be found in [41, 95, 285], for discretemodels in [264, 285] and the following section of our book.

10.6 Chemical Kinetics Equations and H- Theorem:Conditions of Chemical Equilibrium

The most general form for complex chemical reactions can be written as equa-tion [308]

dfidt=−

∑(α,β)∈J

αi

(Kβα f β −Kαβ f α

), i= 1, . . . ,n. (10.6.1)

Here, by f α , we denote a product f α = f1 α1 f2 α2 × ·· ·× fn αn , summation goesthrough finite symmetric multy-index set (α,β), α = (α1,α2, . . . ,αn), and β =

(β1,β2, . . . ,βn). α’s and β’s are nonnegative integers. An addend (α,β) representselementary chemical reaction

α1S1+α2S2+ ·· ·+αnSn

Kαβ�

Kβα

β1S1+β2S2+ ·· ·+βnSn, (α,β) ∈ J,

where Si denote symbols of reacting substances, Kαβ—coefficients of chemical reactionspeeds. Coefficients αi, βi are also known as stoichiometric ratios. Without loss ofgenerality, we consider set J to be symmetrical with respect to permutations α andβ. Moreover, some ratios (α,β) represent null cross-sections: Kαβ = 0. On the otherhand, probably Kαβ > 0, which means the irreversibility of reactions. For example,inside chemical reaction

2H2+O2 = 2H2O,

coefficients are equal to α = (2,1,0), β = (0,0,2), and pair is symmetric.

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Example 10.1. Chemical reaction Michaelis-Menten [213] chemical reaction isdefined by two reactions:

S + EK1�

K−1Q, Q

K2�

K−2P + E,

Here S denotes source component reacting with ferment E and providing ferment-component complex Q. Q itself decays reversibly into product P and the same fer-ment E.

This chemical reaction is represented by the following system of differen-tial equations:

ds

dt=−v1,

de

dt=−v1+ v2,

dq

dt= v1− v2,

dp

dt= v2.

Quantities s, e represent concentration of component S and ferment E; v1 = K1se−K−1q, v2 = K2q−K−2ep the differences in the speed of forward and backwardreactions.

While composing such kind of reactions, one uses the “law of mass action,” rep-resented by system (10.6.1). It reads as follows: The rate of a chemical reaction isdirectly proportional to the molecular concentrations of the reacting substances withrespect to their stoichiometric ratios. This law was established by Guldberg and Waagein 1864–67 (see [175] for details).

In foundations of chemical kinetics and for any statistical interactions, the firstquestion to answer is the existence of H- function. In other words, we would liketo know when system (10.6.1) possesses entropy—type functional decreasing overnonstationary solutions. And how it could be interpreted in chemistry.

The simplest case when functional H =∑

i fi(ln fi− 1) decreases with respect to

equation (10.6.1) is symmetrical, when Kαβ = Kβα for all reactions (α,β) ∈ J. Usually,this family is just called symmetrical S.

Another important case refers to reactions with detailed balance. It merely meansthat we suppose that, for our m reactions, there exists at least one solution of the system

Kαβ f α = Kβα f β , (10.6.2)

also written as

f α1−β11 · f α2−β2

2 · · · f αn−βnn =

KβαKαβ. (10.6.3)

Sometimes, equility (10.6.3) is called law of mass action [170]. We denote as Dapplying logarithmic transformation, we obtain the relation on ln fi, i= 1, . . . ,n. Fam-ily of systems (reactions) fulfilling condition of detailed balance. Solution propertiescan be described by H- function

H =∑

i

fi

(ln

fiξi− 1

). (10.6.4)

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Here ξ is one of the positive solutions of the system (10.6.2). For the detailed descrip-tion see [308] and [318]. Example 10.1 also belongs to D, but, as it will be shownlater, even linear systems posessing H- function do not belong to D. Hence, we needto introduce some appropriate conditions for H- function and system (10.6.1) (com-pare with attempts in [122, 219]).

Condition of dynamical balance (Stukelberg-Batisheva-Pirogov condition).Assume there exists at least one positive solution ξ of the following equation:∑

β

K αβ · ξ

α=

∑β

Kβα · ξβ . (10.6.5)

Parameter α used in (10.6.5) is chosen in such a manner that, for some β, we have(α,β) ∈ J. Summation goes for β : Kαβ 6= 0 or Kβα 6= 0. These systems we denote asE , or dynamically balanced systems.

Theorem 10.1. Let coefficients Kαβ in system (10.6.1) provide at least one solution ξof the equation (10.6.5). Hence,

a) H- function (10.6.4) does not increase along the solutions of the system (10.6.1), i.e.,dH/dt ≤ 0.

b) System (10.6.1) possesses n− r conservation laws of the form∑µk

i fi(t)= Ak= Const,

k = 1, . . . ,n− r, where r is the dimension of the linear hull α−β, vectors µk orthogonalto all α−β :

∑µk

i (αi−βi)= 0. Fixing all constants Ak in conservation laws, we can findunique stationary solution (10.6.1) of the form

f0i = ξie

∑k

µki λ

k

, (10.6.6)

where λk uniquely depends on Ak.c) Unique stationary solution (10.6.6) exists if Ak is defined from initial condition f (0), Ak

=∑µk

i fi(0). Solution f (t) with introduced initial condition exists ∀t > 0, is unique and tendsto stationary solution (10.6.6).

Proof. We proceed, proving all three assertions one by one.

a) We have to study properties of H- function (10.6.4). Deriving and transforming dH/dt, weobtain:

dH

dt=−

∑(α,β)∈J

Kβα ξβyβ lnyβ−α =

=−

∑(α,β)∈J

Kβα ξβyα

[yβ−α lnyβ−α − yβ−α + 1

]≤ 0, (10.6.7)

where yi = fi/ξi. To obtain this relation, we exchanged once α and β in one addend, thenused condition (10.6.5) adding zero component. Inequility (10.6.7) is valid, since u lnu−u+ 1≥ 0. Since u lnu− u+ 1= 0 only for u= 1, then (10.6.7) becomes an equility for

f0 : f α−β0 = ξα−β . (10.6.8)

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Here Kαβ 6= 0 or Kβα 6= 0. Expression (10.6.8) also means that f0 complies with the samecondition (10.6.5) as ξ .

b) One can prove the conservation property of the functional∑µi fi(t) for µ⊥ α−β just by

rewriting (10.6.1) in the following form:

dfidt=−

∑(α,β)∈J

(αi−βi) ·Kαβ · f

α, i= 1, . . . ,n, (10.6.9)

due to simmetricity of the set J with respect to permutations of α and β.Applying logarithimic transformation to (10.6.8), we get that vector with components

of the form ln f0i/ξi is orthogonal to all vectors α−β. Hence, relation (10.6.6) is fulfilledfor all stationary solutions of the system (10.6.1) for some fixed λk. In their turn, coeffi-cients λk are uniquely defined by parameters of the conservation laws Ak obtained from thesystem

∂L

∂λk= Ak, L=

n∑i=1

ξie

n−r∑k

µki λ

k

;

here L is a convex function.c) Since H- function is convex, then a solution of (10.6.1) exists for all positive initial values.

Hence, the set of points f such that H(f )≤ H( f (0)) is a compact set. Due to convexity ofH, there exits a unique point where H gains its minimum for the given set of parametersAk=∑µk

i fi(0) of the conservation law∑µk

i fi(t)= Ak. According to (10.6.7), we have astrict inequility everywhere else, then after a solution tends to the minimum of H- functionin other points.

Finally, we were able to construct the following classification for the equation ofchemical kinetics (10.6.1) based on the entropy principle

S ⊂D ⊂ E ⊂ C.

As C here, we denote the whole class where (10.6.1) is defined for a final number ofreactions. E represents systems with positive solution of the equation (10.6.5), whichmeans that entropy increases.D is the systems (10.6.1) with detailed balance (10.6.2).And the last one S refers to the systems with symmetrical reaction constants whenKαβ = Kβα .

Example 10.2. Linear systems. If set J contains only unit vectors, then (10.6.1)becomes linear and was considered in Chapter 9. Linear system (10.6.1) is also calledthe Pauli master equation. It also is used for studying Markovian processes with finitenumber of states and continuous time. H- theorem, for the linear system (10.6.1), isalready known (see [217] for example). But the Theorem 10.1 proven here also isusefull, since it outlines the role of the conservation laws and describes the set ofstationary solutions for highly sparse matrices.

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As a matter of fact, a certain subclass, namely, linear systems with detailed balance,has several special applications in physics.

Example 10.3. Coagulation—fragmentation equations. Coagulation—fragmen-tation processes can be described via special equations

df (m)

dt=

1

2

m−1∑i=1

[K(l,m− l)f (l)f (m− l)−F(l,m− l)f (m)]−

−

∞∑i=1

[K(l,m)f (l)f (m)−F(l,m)f (m+ l)] .

Here we denote by f (m) a numerical density of particles consisting from m monomericunits, K(l,m) is a kernel (constant) of coagulation, F(l,m)—fragmentation kernel. Anexact relation between the kernels arises from the condition of detailed balance

F(l,m)=K(l,m)ξ(l)ξ(m)

ξ(l+m)

for some positive sequence ξ(l), l= 1,2, . . . This relation was studied in [65, 66].

10.6.1 Conclusion

The most general requirement for discrete models becomes the conservation of theproperties studied for initial equations. The most important is conservativity, or justexistence of the conservation laws [117, 251]. The principle of full or complete con-servativity means that all conservation laws are fulfilled (see e.g. [251]). A naturalquestion arises what does the word “all” mean? For the Boltzmann-type equations andtheir generalizations the linear conservation laws are widely used. This is natural, sincethey are used to define continuous medium equations. As a result, while translatingthem into discrete form, we should reject spurious invariants. This issue is discussedin Chapter 12. One can call this a principle of precise conservativity. This situation istypical for all scientists using computations for kinetic equations [11, 123, 254, 316].

From the theoretical point of view, the situation is clarified by the theorem on theuniqueness of the Boltzmann H- function [285]. From it follows in particular that forthe Boltzmann equation among conservational laws there are only 5 linear standardadditive invariants:

G[g]=∫ϕ05g(v)dv.

Rykov (see [250]) attempted to expand the class of functionals, considering time-dependent functionals of the form∫

ϕ(x− vt,v, f )dvdx.

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Here ϕ(·) is a function of seven variables. It generates eight more invariants (Rykov’sinvariants). For the Broadwell model, their number is even higher and depends onarbitrary functions [10].

Another fundamental question is the correspondence of hydrodynamical equationsderived from the Boltzmann equation and the results of physical experiments.

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11 Method of Spherical Harmonicsand Relaxation of Maxwellian Gas

In this chapter, we will solve Maxwell problem involving the derivation of momentsystem from a Boltzmann equation for Maxwellian molecules. This is based on thegroup symmetries of the Boltzmann collision integral, which has general mathematicalnature. For example, if linear operator commutes with rotation in three-dimensionalspace, then it is multiple to unit operator. Two quadratic operators defined in 3D spaceand commuting with rotation group are well known—they are scalar and vector prod-ucts. Do any others exist? The answer will be revealed at the end of the Section 11.2.It will be shown that there exists only one more operator of such kind, transformingthree-dimensional space into five-dimensional one.

11.1 Linear Operators Commuting with Rotation Group

Let C(S2)—space of complex continuous functions on the sphere S2 and let L—linearoperator transforming C(S2) into itself:

L : C(S2)−→ C(S2).

Let L commutes with rotations Tg, g ∈ SO(3): TgL= LTg, where(Tg f

)(S)=

f(g−1S

), S ∈ S2—vector of two-dimensional sphere; g—an arbitrary rotation; and

SO(3)—orthogonal group with determinant equal to unit.Then L possesses the following properties [156]:

1. spherical harmonics Ylm are eigenfunctions of operator L:

L(Ylm)= λlYlm; (11.1.1)

2. eigenvalues λl do not depend on number m.

A similar situation occurs quite often in applications. As an example we refer to anangular part of Laplace operator.

Such formulas are connected with the fact, that spherical harmonics Ylm(m=−l, . . . l) generate basis of irreducible representation of rotation group (dimen-sion of representation space equals to 2l+ 1).


http://dx.doi.org/10.1016/B978-0-12-387779-6.00011-9

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Definition 11.1. Let G—group. A couple (H,T) is called a representation of groupG, where H is a linear space, if T= T(g) is a collection of linear operators from Honto H depending from elements g ∈G. Here mapping g→ T(g) is homomorphismof the group G into the group of linear invertible operators L(H) of the space H. Inother words, T is a homomorphism G in L(H): G→ L(H).

Representation is called irreducible, if there are no linear subspaces in H invariantover all operators T(g) different from zero and H.

Independence of eigenvalue λl from m is formulated in Shur’s Lemma [156].

Lemma 11.1. Let G is a group, (T,H) is some irreducible representation of group G,L is a linear operator, commuting with all operators T(g). Then L is multiple to unitoperator: L= λE.

Exercise 11.1. Prove Shur’s Lemma.

Applying Shur’s Lemma to the rotation group and using that Ylm basis of irre-ducible representation, we obtain formulas (11.1.1) and independence of λl fromm. We will call this representation (Yl,Tl) according to the definition, where Yl is(2l+C)-dimensional space with basis Ylm.

11.2 Bilinear Operators Commuting with Rotation Group

Here we apply representation theory to quadratic operators B commuting with rotationgroup. Collision integral with arbitrary central interaction law between particles issuch an operator. Let B be bilinear operator on the space of continuous functions onthe sphere S2:

B : C(S2)×C(S2)→ C(S2).

Assume also that B commutes with rotations:

B(Tg f ,Tgh)= TgB(f ,h).

First of all, let us see which spherical harmonics represent the decomposition ofB(Yl1m1 ,Yl2m2

). The answer can be found in [108, 119, 156, 172, 285]:

B(Yl1m1 ,Yl2m2

)=

∑l

⟨l1 l2 lm1 m2 m1+m2

⟩Bl

l1l2Yl,m1+m2 . (11.2.1)

This means that summation takes place only by finite number of spherical harmon-ics with m= m1+m2, |l1− l2| ≤ l≤ l1+ l2. All dependences from m are contained

in Klebsh-Gordon coefficients

⟨l1 l2 lm1 m2 m

⟩, and information about operators is con-

tained in coefficients Bll1l2

independent of m. This values sometimes are called bilin-ear eigenvalues since they are eigenvalues numbers of some linear operator connectedwith B.

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Method of Spherical Harmonics and Relaxation of Maxwellian Gas 197

Taking into account that mentioned formulas represent general situation, the proofwill be presented in terms of representation theory for simplicity. Let (Hi,Ti), i=1,2,3—three representations of some group G, the first two of them are irreducible.Let the direct product H1×H2 transforms to H3 by bilinear operator B, commutingwith G. We continue B up to linear operator B from tensor product H1⊗H2 to H3.

Here we want to remind the difference between direct H1×H2 and tensor H1⊗H2products. Let ei—basis in H1 and fj—basis in H2. Then the basis of H1×H2is the ei = (ei,0) and fj = (0, fj), therefore dimension dim(H1×H2) is equal tothe sum of respective dimensions dimH1+ dimH2. The basis of H1⊗H2 areei⊗ fj and dimension dim(H1⊗H2)= dimH1 dimH2—i.e. the product of initialdimensions.

Continuation is constructed by formulas B(ei⊗ fj

)= B

(ei, fj

)and is continued by

linearity. Additionally in H1⊗H2 tensor product T1⊗T2 of representations T1 andT2 is defined.

Hence it leads to the problem how to decompose tensor product T1⊗T2 of repre-sentations onto irreducible representations. The decomposition coefficients are calledKlebsh-Gordon coefficients. For spherical harmonics, we obtain

Yl1m1 ⊗Yl2m2 =

∑|l1−l2|≤l≤l1+l2

⟨l1 l2 lm1 m2 m1+m2

⟩Yl,m. (11.2.2)

Operator B is linear and acts onto Yl,m by formulas B(Yl,m

)= Bl

l1l2Yl,m. Depen-

dence of numbers Bll1l2

from l1 and l2 is explained by restriction of the operator B onYl1 ⊗Yl2 .

We note that summing in formulas (11.2.1) and (11.2.2) is conducted by the samel found in the theorem on summing of angular momentum in quantum mechanics. Anexpression (11.2.1) also contains an answer for the question stated in the beginningof the chapter. Since for three-dimensional representation space we put l1 = l2 = 3,then summation in (11.2.1) and (11.2.2) becomes l= 0,1,2. This corresponds to theequality 9= 1+ 3+ 5 in terms of dimensions. The scalar product represents a trace ofmatrix 3× 3 of multiplications aiaj. Vector product is a skew-symmetric part. Anotheroperator is one in five-dimensional space—traceless part of symmetrization, given by

a matrix:aiaj+ ajai

2−(a11a11+ a2a2+ a3a3)

3.

11.2.1 Computations of Eigenvalues

We can compute λl in (11.1.1) using Yl0, which is proportional to Legendre polynomialPl. Substituting it in formulas (11.1.1) and using equality Pl(1)= 1, one obtain

λl = L(Pl (cosθ))(0,0,1).

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We can calculate Bll1,l2

using orthogonality condition for Klebsh-Gordoncoefficients: ∑

(−l1,−l2)≤(m1,m2)≤(l1,l2)

⟨l1 l2 lm1 m2 m

⟩⟨l1 l2 l′

m1 m2 m′

⟩= δll′δmm′ .

Hence we obtain

Bll1l2=

(4π

2l+ 1

) 12 ∑−max(l1,l2)≤m≤max(l1,l2)

⟨l1 l2 l−m m 0

⟩×

×B(Yl1,−m,Yl2,m

)(0,0,1).

Next, we apply these reasons for a Boltzmann equation. First, we perform a Fouriertransformation of collision integral by velocity to simplify further calculations. Thistechniques was proposed by A. B. Bobylev [41] and is known as the simplificationmethod of collision integral.

11.2.2 Fourier Transformation of Collision Integral

Let us consider space-uniform Boltzmann equation in the form

∂f1∂t= I[f1, f2]=

∫dwdn

[f1(v′)f2(w

′)− f1(v)f2(w)]

B

(|u|,

(u,n)

|u|

). (11.2.3)

Multiplying this equation by e−ikv and integrating over dv we obtain new equationfor

ϕi(k)=∫

fi(v)e−i(k,v)dv,

namely

∂ϕ1

∂t= S(ϕ1,ϕ2)=

∫I [f1, f2]e−i(k,v)dv.

Our task is to obtain an expression depending on ϕ1 and ϕ2 only.We transform this integral in the following manner:

S(ϕ1,ϕ2)=

∫f1(v)f2(w)

[e−i(k,v′)

− e−i(k,v)]

Bdwdvdn.

Here we used by property of changes (v,w)→(v′,w′

)from Section 9.2.

Taking common multiplier e−i(k, v+w2 ), we obtain

S(ϕ1,ϕ2)=

∫dvdwf1(v)f2(w)e

−i(k, v+w2 )F(u,k)= S+− S−, (11.2.4)

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where

F(u,k)=∫

dn

e−

i(k,n)n

2 − e−

i(k,u)

2

B

(|u|,

(u,n)

2

)

and B(u,s)=∫

ei(p,u)ψ(p,s)dp.

According to this equation obtain F(u,k)=∫

dpei(p,u)G(k,u, |p|), where

G(k,u, |p|)=∫ [

exp

(−

i(k,n)

2|u|

)− exp

(−

i(k,u)

2

)]ψ

(|p| ,

(u,n)

|u|

)dn.

Exercise 11.2. Prove that the function G(k,u, |p|) is symmetric with respect to argu-ments k and u: G(k,u, |p|)= G(u,k, |p|).

Proof. If we rotate vectors k and u by the same angle, then function G does not change.It means that it depends only from |k|, |u| and (k,u). Therefore it depends symmetri-cally from |k| and |u|.

So, trading k and u in places, one obtain expression for F

F(u,k)=∫

dndp

ei(p,u)− i|k|

(u,n)

2 − e−i(p,u)−i (u,n)2

ψ(p,(n,k)

|k|

).

(11.2.5)

Collecting terms with v and w in exponent, in term S+ from (11.2.4) and (11.2.5)one obtain for v

e−i(v,k)

2+ i(p,v)− i|k|(n,v)

= e−i

(v,

k+ |k|n

2− p

)

and for w

e−i(w,k)

2− i(p,w)+ i|k|(n,w)

= e−i

(w,

k− |k|n

2+ p

).

Under the same assumptions in term S− one obtain:

e−i (k,v)2 + i(p,v)−i (k,v)2 = e−i(v,k−p),

e−i (k,w)2 −i(p,w)+ i (k,w)2 = e−i(w,p).

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Integration over dv and dw gives

S(ϕ1,ϕ2)=

∫dndpψ

(|p|,

(k,n)

|k|

)×

×

[ϕ1

(k+ |k|n

2− p

)ϕ2

(k− |k|n

2+ p

)−ϕ1(k− p)ϕ2(p)

].

The utmost simplification is reached if B(|u|,s) does not depend on |u|—a termrepresenting Maxwellian molecules. Then 9(p,S)= δ(p)g(S) and we obtain [41]

S (ϕ1,ϕ2)=

∫dn dpψ

(|p|,

(k,n)

|k|

)× (11.2.6)

×

[ϕ1

(k+ |k|n

2

)ϕ2

(k− |k|n

2

)−ϕ1(k− p)ϕ2(p)

]. (11.2.7)

11.3 Momentum System and Maxwellian Gas Relaxation toEquilibrium. Bobylev Symmetry

We obtained Boltzmann equation for Maxwellian molecules when distribution func-tion f (v, t) is independs from space variable in Fourier representation:

∂ϕ

∂t= S(ϕ,ϕ), (11.3.1)

where integral S is defined by formulas (11.2.6).To study the relaxation, we assume:

ϕ = ϕ0(1+ h), (11.3.2)

where ϕ0 = e−k2

2 —Fourier image of Maxwellian distribution, h—deviation ofMaxwellian distribution.

Parameter h is described by the following equation:

∂h

∂t= L(h)+ S(h,h), (11.3.3)

where L is the linearized collision operator in Fourier representation:

L(h)=∫ [

h

(k+ |k|n

2

)+ h

(k− |k|n

2

)− h(k)− h(0)

]×

× g

((k,n)

|k|

)dn. (11.3.4)

Exercise 11.3. Obtain (11.3.3) from (11.3.1) taking (11.3.2).

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Both operators: L and S commute with rotation group. This is a general propertyfor all collision integrals. The critical property for Maxwellian molecules refere to therelation with one more Abelian group: convolution of the solution with Maxwelliandistribution also makes a solution. Fourier representation for the last one is essentiallysimple: both operators commute with dilation group [41] also. This is the Bobylevsymmetry:

L (fα)(k)= L(f )(αk), where fα(k)= f (αk).

The same goes for S.Thanks to this property, |k|rYlm are eigenfunctions of operator L [41]:

L

(|k|rYlm

(k

|k|

))= λrl|k|

rYlm

(k

|k|

).

Dilations possess eigenfunctions |k|r, and spherical harmonics are connected withrotations. Using formulas from section 11.2.1 we obtain the following expression foreigenvalues

λrl =

∫ [cosr θ

2Pl

(cos

θ

2

)+ sinr θ

2Pl

(cos

θ

2

)− 1− δr0δl0

]g(cosθ)sinθdθ.

(11.3.5)

For the bilinear operator, one have:

S(|k|r1Yl1m1 , |k|

r2Yl2m2

)= |k|r1+r2×

×

∑|l1−l2|≤l≤l1+l2

⟨l1 l2 lm1 m2 m1+m2

⟩Yl,m1+m2

(k

|k|

)3

r1r2ll1l2

.

Here an exact values for 3r1r2ll1l2

are similar to (11.3.5), see [285]. On the basis of theseformulas, one can obtain momentum system for Boltzmann equation. Let

h=∑rlm

Crlm(t)|k|rYlm

(k

|k|

).

Such momentum system corresponds to decomposition of distribution function byLaggere polynomials multiplicated by spherical harmonics:

f (v, t)=∑|v|lL

l+12

r−l2

(v2

2

)Ylm

(v

|v|

)Mrlm(t).

Hence for the momentum Crlm(t) we obtain the following system of equations:

Crlm(t)= λrlCrlm+ψrlm(C),

ψrlm(C)=∑

r1l1l2m1

⟨l1 l2 lm1 m−m1 m

⟩Cr1l1m1Cr−r1,l2,m−m13

r1,r−r1,ll1,l2

.

A special feature of the obtained system is that ψrlm includes momentum of lowestindex r and, therefore, it is recursively solvable.

To be exact, that fact made it possible for Maxwell to solve the problem, deriv-ing the equation of continuous medium from a Boltzmann equation. However,

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Chapman-Enskog solution for arbitrary potentials is essentially based on givensolution for Maxwellian molecules. For Maxwell-Chapman-Enskog method, it is suf-ficient deal with first momentum equations. The final solution of momentum systemwas been obtained quite recently, see [41, 285].

11.4 Exponential Series and Superposition ofTravelling Waves

A lot of papers devoted to the study of scattering theory (and, in particular, to theinverse problem method) revealed a lot of equations in which one can obtain anexplicit analytical expressions for superposition of travelling waves [317] (n-solitonsolutions). Analytical methods introducing wave superposition as series [39–41, 46,94, 190, 204, 243, 283, 285] were also proposed. Papers [40, 46, 243] establishedcorrespondence between the series and exact solutions for the Korteweg-de Vriestype equations. But this idea cannot be used freely due to existence of special rela-tions between eigenvalues, named as resonance. In [40, 46, 243] it was provedthat resonances do not arise in KdV-type equations, followed by similar results fornonlinear elliptic equations [94, 190] and isotropic Boltzmann equation [39]–[41].Papers [283, 285] discovered this fact for anisotropic Boltzmann equation. Specialefforts were made in papers [94, 190] studying convergence of this series at small(i.e. in some neighborhood of zer) for nonlinear elliptic equation. On contrast, in [39–41, 204, 283, 285] such series were constructed for Boltzmann equation and therewere proved convergence theorems in general (i.e. in the hole complex space Cn, n isa number of waves).

In this section our attention is paid mainly to evolutionary equations (11.4.1).In the next subsection, 11.4.2, we introduce formal decompositions in exponentialseries—series representing interactions of traveling waves. Subsection 11.4.3 detectsexisting resonance relations. In Subsection 11.4.4, we introduced several convergencetheorems at small and provided some divergence examples. Here under the name ofconvergence at small we understand that a function representing a superposition oftravelling waves (called sometimes an interaction function) gives a solution of corre-sponding initial equation in some small domain. In article [155] and later publicationsof its author numerically was shown that for hyperbolic nonlinear equation corre-sponding series are fast converging on the whole plane, it seems that they do not payattention to existing resonaces.

The resonance study and convergence study are of the highest importance, sincethey provide a background for numerical methods solving the problem of travelingwaves interaction for any nonlinear equation with coefficients independent of spaceand for any dimension.

11.4.1 Equations of the Form ∂u/∂t = F(u)

Considering equations of the Form

∂u

∂t= F(u), (11.4.1)

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for u(x, t) and x ∈ R, t ∈ R; F—nonlinear operator commuting with a shift group g(a) :x→ x+ a, [F,g(a)]= 0. Let F(u)= L(u)+B(u)+ ·· · where L(u) is linear, B(u) isbilinear and the rest represent operators commuting with a shift group. Operator F ofthis type includes all differential equations where partial derivatives over x have con-stant coefficients. Our interest is focused at the second order equations, since higherorder ones are studied in similar manner. Main examples are the ∂u/∂t = u(m)+ uxuequation (for m= 2 it is called Burger’s equation, for m= 3 it is a KdV equation).Also we will discuss a generalised equation ∂u/∂t = u(m)+ u(p)u(q).

11.4.2 Waves Interaction Series

Let vectors α = (α1, . . . ,αn), β, γ are real or complex valued vectors. Also suppose

u=∑

dke(k,αx+βt+γ ). (11.4.2)

Summation goes by vectors k with nonnegative integer components; at least one com-ponent is different from zero. If we substitute (11.4.2) into (11.4.1) the resulting rela-tion allows to define coefficients dk consequently:

[(k,β)− λ((k,α))]dk =∑

0<r<k

3((r,α),(k− r,α))drdk−r (11.4.3)

Summation goes over vectors r, where r< k means that inequality less or equal holdsfor all but one vector component, which have to be strictly less.

Coefficients λ(p) and 3(p,q) are defined from expressions

L(epx)= λ(p)epx,

B(epx,eqx)

=3(p,q)e(p+q)x.

Regarding system (11.4.3) assume |k| = k1+ ·· ·+ kn. If |k| = 1 i.e., k= ej thenexpression (11.4.3) for dej 6= 0 becomes

βj = λ(αj). (11.4.4)

When |k|> 1, coefficients dk are defined consequently from (11.4.3), if for βj =

λ(αj) multiplyer (k,β)− λ((k,α)) 6= 0. In other words, this differences, also calledresonances

λ((k,α))−∑

kjλ(αj)= 0 (11.4.5)

are obstacles for construction solutions of the form (11.4.2).When n= 1 in (11.4.2), vectors α,β,γ and k become numbers and our solution

takes the form of traveling wave

u= f (x+ ct)=∑

1≤k≤∞

dkekα(x+ct+γ ).

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Definition 11.2. Define λ(p) to guarantee that for some vector α ∈ Cm and ∀k withnonnegative integer components, |k|> 1 relation (11.4.5) is not fulfilled. Then weintroduce V as a formal interaction series (nonlinear summation, overlapping) ofn travelling waves for the equation (11.4.1)

Vn = V(F;n;α;z)= Vαn (z)= Vn(z1,z2, . . . ,zn)=∑

dkzk11 . . .z

knn . (11.4.6)

Here dk = 1 for |k| = 1. When |k|> 1, coefficients dk are defined by expressions(11.4.3), (11.4.4). Summation in (11.4.6) is defined similarly to earlier cases: overnonnegative integer vectors k : |k| = 1. Variable z connecting expressions (11.4.2)and (11.4.6) is defined by relation zi = eαix+βit+γi .

Our goal is to verify the existence of resonance ratios (11.4.5) and convergence ofseries (11.4.6).

11.4.3 Investigating Resonance Relations

Assume α = (α1, . . . ,αn) a real vector with positive components.

Lemma 11.2. Let λ(p)—polynom of the second or higher degree. Then condition(11.4.5) is not fulfilled for large |k|.

Proof. Let assume 0< α1 ≤ . . .≤ αn. Hence for r ≥ 1

|k|rαrn ≥

(∑kjαj

)r≥ |k|rαr

1.

Comparing the sign of the difference in (11.4.5) and the sign of the coefficient athighest degree of λ(p) for |k| →∞:

if λ(p)= a0+ a1p+ ·· ·+ adpd (d ≥ 2, ad 6= 0),

then λ((k,α))−∑

kjλ(αj)= ad

(∑kjαj

)d+ o

(|k|d

),

where o(x)→ 0 for x→∞.

Lemma 11.3. Let λ(p)—polynom of the second or higher degree with positive coeffi-cients and non zero free coefficient. Then λ((k,α)) >

∑kjλ(αj) for |k|> 1.

Proof. Taking monomial λ(p)= pr with r ≥ 1 and |k| ≥ 1 we have following inequa-lities

λ((k,α))=(∑

kjαj

)r≥

∑kr

jαrj ≥

∑kjα

rj =

∑kjλ(αj).

Let r > 1 and |k|> 1. Then if vector k has several components different from zero,then the first inequality is strict. If we have only one component different from zero,the second inequality is strict also. Calculating the sum of inequalities for different rwith nonnegative coefficients, we finish the proof.

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Lemma 11.4. Let λ(p)—polynom of the second or higher degree with complex coeffi-cients and non zero free coefficient. If all non zero coefficients are on the same side ofcertain line passing through zero point, then resonances (11.4.5) do not exist.

Proof. Let λ(p)= a0+ a1p+ ·· ·+ adpd, d ≥ 2, ad 6= 0 for ar = br + icr. Due tolemma conditions Abr +Bcr > 0 if ar 6= 0 where Ax+By= 0 if an equation of linepassing through zero point. Using inequality proved in Lemma 11.3 for λ(p)= pr

we make summation for r with nonnegative coefficients Abr +Bcr. Also taking intoaccount that for r = d inequality is strict, we finally have

<(A− iB)λ(k,α)=∑

(Abr +Bcr)(k,α)r >

>∑

(Abr +Bcr)(∑

kj,αrj

)=<(A− iB)

∑kjλ(αj).

This inequality contradicts condition (11.4.5).

Theorem 11.1. Under conditions established in Lemma’s 11.3 or 11.4 solutions(11.4.2) of the equation (11.4.1) exist ∀α ∈ Rn with positive components and ∀γ ∈ Cn.In the case of Lemma 11.2 we can guarantee the same slightly varying coefficients ofoperator L.

11.4.4 Convergence of the Series of n Interacting Traveling Waves

The following theorem holds:

Theorem 11.2. Let for equation ∂u/∂t = L(u)+B(u,u)

1. resonance relations (11.4.5) are absent;2. there exist such constant C = C(n,α) > 0 that for some α ∈ Rn and ∀k : |k|> 1 holds an

inequality

supr|3[(r,α),(k− r,α)]|∣∣λ((k,α)−∑kjλ(αj))

∣∣ < C. (11.4.7)

Hence function Vαn (z) for superposition of n travelling waves is defined by a conver-gent series (11.4.6) in the neighborhood of zero.

Proof. By induction over k we can prove that for k≥ 1 hold an inequality∣∣∣dk∣∣∣≤ C|k|−1 (k1+ ·· ·+ kn)!

k1!k2! · · ·kn!. (11.4.8)

It is correct for k= 1 since de(j) = 1. To continue with the proof, we need the followingidentity for polynomial coefficients

∑ (m1+ ·· ·+mn)!(r1+ ·· ·+ rn)!

m1!m2! · . . . ·mnr1!r2! · . . . · rn=(k1+ ·· ·+ kn)!

k1!k2! · . . . · kn

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where summations go for vectors with nonnegative integer components r,m : r+m= k. Supposing that inequality (11.4.8) is already proved for smaller k, then forthe next dk we have

|dk| ≤∑ |k− r|!Ck−r−1

|r|!Cr

|k1− r1|!|k2− r2|! · . . . · |kn− rn|!|r1|!|r2|! · . . . · |rn|!=

=k!Ck−1

|k1|!|k2|! · . . . · |kn|!.

With respect to estimation (11.4.8) an absolute convergence of the Taylor series forfunction Vn follows immediately from polynomial formula and properties of geomet-ric progression in the neighbourghood of zero.

As an example, we will establish some results.

Theorem 11.3. Considering equation

∂u

∂t= u(m)+ u(p)u(q). (11.4.9)

we have

1. if m≥ p+ q, then series Vαn defining superposition of n travelling waves converges in theneighbourhood of zero for vectors α with positive components;

2. the same series diverges (even for n= 1) if max(p,q)≥ m+ 1.

Proof. 1) In this case resonances are absent: see Lemma 11.3. Then

|3[(r,α),(k− r,α)]|∣∣λ(k,α)−∑kjλ(αj)∣∣ = |[(r,α)p(k− r,α)q]|∣∣∣(k,α)m−∑kjα

mj

∣∣∣ ≤≤

ppqq

(p+ q)p+q

(k,α)(p+q)∣∣∣(k,α)m−∑kjαmj

∣∣∣ . (11.4.10)

The inequality is obtained by ordinary differentiating and searching for max-imum of the function (r,α)p(k− r,α)q over (r,α) for 0< r< k. It equals

ppqq

(p+ q)p+q(k,α)(p+q). For m≥ p+ q the right side of inequality (11.4.10) is bounded

by constant C(α) and thus shows the applicability of the Theorem 11.2.2) first of all, consider equation ∂u/∂t = u(m)− u(p)u(q) which differs from (11.4.9)only in sign. But this guarantee ∀dk > 0. Moreover, taking max(p,q) > m, m≥ 2series Vα1 corresponding to this equation has

dk > bkk! (11.4.11)

for some b> 0 for n= 1 and α > 0 (here α ∈ R+ and k ∈ Z+).

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Now assume p≥ q and p≥ m+ 1. The proof of (11.4.11) will by done by inductionover k. Starting from k = 1 we have b< 1. Now we choose 0< b< 1 sufficientlysmall to fulfill the following inequality

αp+q+m(k− 1)p

k2(km−1− 1)≥ b (11.4.12)

for all k ≥ 2. It can be done for p≥ m+ 1.From recurrent relation for dk we obtain

dk =αp+q+m

k(km−1− 1)

∑rp(k− r)qdrdk−r ≥

≥αp+q+m(k− 1)p

k2(km−1− 1)· kdk−1 ≥ bkdr−1 ≥ bkk!.

To obtain the first inequality we reduced the expression keeping the only summandwith r = 1 (since all another summands are positive). For the second inequality weused (11.4.12). The last one is supposed to be fulfilled due to reasoning by induction.

Series∑

bkk!zk has a zero convergence radius ∀b. Hence the convergence radiusof (11.4.6) also equals zero. To be able to obtain equation (11.4.9) we just note thatdk’s alternate their signs and differ from calculated only in their sign for even k. As itfollows, convergence radius of this series for (11.4.9) also is zero.

Theorem 11.4. Regarding equation

ut = L(u)+ f(

u,u′, . . . ,u(m))

(11.4.13)

where L is a linear operator L(u)= λ(∂/∂x)u and λ is a polynomial

λ(u)=M∑

i=0

aiui

we assume

1. resonances (11.4.5) do not exist for operator L;2. f (0)= 0, Of (0)= 0, function f is analytical at zero;3. M ≥ m.

Hence series Vαn =∑

dkzk converges in the neighborhood of zero when ai ≥ 1 foreach n.

Proof. Using majoring technique (see [94, 190]) consider an auxiliary equation

aW(m)= f

(W(m), . . . ,W(m)

)+

(∑αm

i zi

)a

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and corresponding series Wαn =

∑wkzk with the same α and n. W(m) is an m-th deriva-

tive over x, zi = eαix. Lets prove that by proper choice of a and f series Wαn can be

made majorant series for series Vαn ; namely, wk ≥ |dk| while Wαn is analytical at zero.

f will be defined by a convergent series f =∑|fk|rk. Now we define a> 0 to fulfill

inequality∣∣∣λ((k,α))−∑kiλ(ai)

∣∣∣≥ a(k,α)m.

It can be done since resonances are absent, M ≥ m and estimations of Lemma 11.2holds. Using series defined as above, we obtain the desired result for series Wα

n andVαn , since

(a) all coefficients of polynom expressing wk in terms of wr, r < k are positive;(b) replacing arguments of function f by higher derivatives, we increase the values of the coef-

ficients since (k,α)p is a monotonic no decreasing function of variable p with αj ≥ 1;(c) choosing parameter a we reduced the values of denominator, i.e. coefficients for Wk caming

from the linear term of equation.

Equation for W(m) is algebraic equation of the form

F(

W(m),y)= 0, y=

∑αm

1 z1.

Here ∂F/∂W(m)= a> 0, then this equation can be solved according to the theorem of

implicit function. Thus W(m) is analytical at zero and can be represented as

Wα(m)=

∑k

wk(k,α)mzk.

Being replaced by Wαn =

∑wkzk it only improves convergence.

11.4.5 Final Remarks

1. For the Burger’s equation ∂u/∂t = uxx+ 2uxu we have

Vαn =z1+ ·· ·+ zn

1+ z1α−11 + ·· ·+ znα

−1n.

In the KdV case functions Vαn a rational by z, but polynomial degrees for numerator anddenominator grow along with n grows [317]. Functions Vαn constructed in Theorems 11.1–11.3 provide us a solution of the equation (11.4.1)

u(x, t)= Vn(eθ1 ,eθ2 , . . . ,eθn), θj = αjx+ λ(αj)t+ γj

inside the convergence domain. Theorems 11.2, 11.3 guarantee convergence ∀αj > 0 andsufficiently small x. In particular, publications [40, 46, 190, 243] proved that n-solution forthe KdV-type equations just coincide with the corresponding (11.4.6)-type solutions.

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2. Theorems 11.1–11.3 also could be considered as analogs of the Poincare-Horn theorem forODEs. This theorem [13, 57, 139] allow to construct particular solutions for the equations

dxj

dt= Fj(x), j= 1, . . . ,m.

Solutions in the form of convergent series in the neighborhood of the zero stationary point,Fj(0)= 0. Assume that linear part of operator L in (11.4.13) has n eigenvalues lying on thesame side of some line passing through zero point. Under this assumption theorem says thatthere exist an analytical substitution translating the original system into the linear one over ndimensional invariant submanifold. Thus generating a set of particular n parameter solutions.

By contrast, Theorem 11.1 acts as an existence theorem of formal series, while Theo-rems 11.2, and 11.3 establish convergence conditions.

3. All established results easily could be extended for multidimensional variable x.

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12 Discrete Boltzmann EquationModels for Mixtures

12.1 Discrete Models with Impulses on the Lattice

Recently the investigations related with study of the discrete models for Boltzmann(DM-BE) were quite active [116], [42], [43] [68], [121], [127], [209], [230], [264].The models of mixtures which allow interchange of energy between different compo-nents are widely discussed in literature. The construction of such models has a lot ofapplications. We read in [43]:

“In fact no model (except trivial ones) has appeared before. The simplest modelsare given by Monaco and Preziosi in their book [209]. They are not satisfactory (asindicated by the authors [209], p. 74), because no exchange of energy between thespecies occurs. It is just what we mean when we say that they are trivial.”

The construction of correct models (at least for the case of two components) isdefined by exceptional difficulty to overcome the restrictions of an additional invari-ants (for example—energy of independent components) [43], [68], [209] which arepresent in discrete model, but absent in initial kinetic equation. This issue results inwrong calculated hydrodynamics if we speak about numerical modeling, since locallyequilibrium distribution in this case is not Maxwellian. Below, we describe the methodallowing to solve this problem effectively and construct discrete models with correctnumbers of invariants (normal models).

Consider a mixture of particles of r kinds with given masses m1, . . . ,mr. OrdinaryBoltzmann equation (with continuous velocities) has the form (see Section 9.3):

∂Fi

∂t+

(pmi,∂Fi

∂x

)=

r∑j=1

Qij[Fi,Fj

], p,x ∈ Rd, (12.1.1)

where Fi(x,p, t)—distribution function of i-th component of mixture by coordinates xand impulses p at moment t. Assuming for simplicity Fi ≡ F, Fj ≡ f , mi ≡M, mj ≡ m,M > m> 0 and keeping impulses as only arguments of distribution functions, colli-sion integral Qij is written

Qij[F, f ]=∫

Rd×Sd−1

dqd�|u|σij

(|u|2,

(u,�)|u|

)×[F(p′)f (q′)−F(p) f (q)

],

(12.1.2)


http://dx.doi.org/10.1016/B978-0-12-387779-6.00012-0

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where u=pM−

qm

—relative velocity; σij—cross-section of collisions, p′ and q′—

impulses of the particles after collision:

p′ =p+q

2(1+ δ)+

1

2|u|µ�, q′ =

p+q2

(1− δ)−1

2|u|µ�,

where µ=2mM

m+M—reduced mass, δ =

M−m

M+m, � ∈ Rd, |�| = 1.

For the particles of one kind the class of Goldstein-Styurtenvant-Broadwell mod-els [121] is used; the velocities are taken on integer lattice. In [42] was proposed thetheorem on approximation of Boltzmann equation in 3D case; in [43] it was givena generalization for the case of mixture. In this book also were proposed two 2DDM-BE models for case of mixture of two kinds and energy exchange. However, thefirst one (with 13 discrete velocities) possesses one extra invariant. The second one(with 25 velocities)—two extra invariants.

If we construct a normal DM-BE over the space of model’s linear invariants whilethe number of discrete impulses increases, we define the basis by the given set J ofvectors α−β and thus eliminate an extra invariants.

Introduce uniform grid in the space of impulses with step size h as follows: the gridnodes correspond to values of impulses pm =m · h, m—vectors with integer coordi-nates. Writing down conservation laws for impulse and energy:

p+q= p′+q′

p2− (p′)2

2M=(q′)2 − q2

2m,

(12.1.3)

we see that allowing exchange of energy between two components, the relation ofmasses M/m should be rational. Without energy exchange, the relation of masses canbe arbitrary and the energies of isolated components are kept.

Formally speaking, discrete model with impulses on the lattice is a collection of:A) masses m1, . . . ,mn (some of them can be equal); B) vectors p1, . . . ,pn and C)reactions (ij)↔ (kl). The last one are the sets of four integer indeces, decom-posed in couples. Denote this set of quadruples ((ij)↔ (kl)) as S. It indicates non-trivial cross-sections of collisions σ ij

kl. For a collection{m1, . . . ,mn;p1, . . . ,pn;S;

σijkl((ij), (kl)) ∈ S

}we define the corresponding system of differential equations, also

called a discrete collision model:

∂fi∂t+

(pi

mi,∂fi∂x

)=

∑σ

ijkl( fk fl− fi fj), i= 1, . . . ,n. (12.1.4)

Summation take place for all quadruples from set S, which involve index i. In otherwords, reactions with σ ij

kl 6= 0. We emphasize the fact, that equation (12.1.4) is veryconvenient to operate mixture elements with coinciding masses, since it avoids two-index notations: the real quantity of different masses r is significantly less than n.Here r = 2.

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Discrete Boltzmann Equation Models for Mixtures 213

A model of collisions described above is called a discrete model of the Boltzmannequation, if and only if the reactions defined comply with conservation laws forimpulse and energy:

pi+ pj = pk+ pl,

pi

2mi+

pj

2mj=

pk

2mk+

pl

2ml,

(12.1.5)

all impulses are squared in the second expression. There are r+ d+ 1 invariants forBoltzmann equation (11.3.4) corresponding to conservation of r kinds of particles, dcomponents of impulse and integral of total energy. Following [43], [68] such DM-BEwith r+ d+ 1 invariants will be called normal.

Functional

I =r∑

i=1

µi fi (12.1.6)

is unnecessary invariant here, if it is conserved by the discrete model, but it is not oneof these r+ d+ 1 invariants, i.e., it is not presented in the form of linear combinationof invariants of Boltzmann equation.

12.2 Invariants

Let us study when functional I =∑µi fi will be conserved according to the (12.1.4).

For space-uniform version of equation (12.1.4) we have

dI

dt=

∑µiσ

ijkl( fk fl− fi fj)=

1

2

∑σ

ijkl fk fl(µk+µl−µi−µj).

We see that for all collisions from S condition µi+µj = µk+µl is necessaryand sufficient condition for I to be an integral. Condition I =

∑ri=1µi fi is integral

of equation (12.1.4) is equivalent to orthogonality condition of vector µ to the vectorseij

kl = ei+ ej− ek− el (ei—standard basis vector), which has σ ijkl > 0.

Denote J—linear hull of vectors eijkl, J⊥—an orthogonal complement (space of

invariants), d(J) and d(J⊥)—their dimensions respectively: d(J⊥)= n− d(J). Vec-tors eij

kl also will be called collision vectors.

Example 12.1. Two-dimensional model with six velocities situated in the corners ofproper hexagon.

In that case r+ d+ 1= 1+ 2+ 1= 4. However, for the given model modulesof velocities are equal to unit, hence, the number of particles coincides with energyand the correct number of invariants is lowered up to 3. Number of collisionsequal to three, but due to existing relation between them d(J)= 2. Therefore,d(J⊥)= 6− 2= 4 and we have one (by the way, it is well known) extra invariant.

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Namely, such additional invariants exist for more general models with velocities inthe corners of the proper 2n—gons [264].

Exercise 12.1. Calculate the number of extra invariants for 2n—gon.

12.3 Inductive Process

We shall construct the discrete model step by step inserting new particles (addingnew impulse) and collisions from the set S. How the dimensions d(J) and d(J⊥) arechanged?

1. If we add particle with new velocity, then n→ n+ 1, therefore d(J)→ d(J) and we havenew invariant d(J⊥)→ d(J⊥)+ 1.

2. If add collision with vector eijkl and this vector is

a. linear independent from previous ones, then we “kill” one invariant: d(J⊥)→ d(J⊥)− 1;b. linear dependent from previous ones, then we have the same number of invariants:

d(J⊥)→ d(J⊥).

Now we show by induction that two-dimensional model on the square lattice withthe side L for sufficiently large L do not have extra invatiants. This statement hasbeen implicitly used in [42, 43, 121], but perhaps it is incorrect in one-dimensionalcase.

We start with classical Broadwell model for the light component of mixture,when there are four discrete impulses only: p2,4 = (0,±1) and p1,3 = (±1,0). Admis-sible collision between these particles is uniquely defined in our notations: e13

24. Sincea sum of dimensions of space spanning onto collision vectors d(J) and it orthogo-nal complement d(J⊥), corresponding to quantity of linear independent invariantsis equal to number of degrees of freedom of the system n, then in given cased(J⊥)= 4− 1= 3.

Now if we add four heavy particles possessing the same impulses (marked by the

wave over index) to the existing light ones, and introduce collision vector e1324

, thenwe obtain six invariants (three for light and three for heavy particles). Adding col-

lision vectors between these two kinds of particles e1324

, e3142

, we reduce the numberof invariants down to four. Finally, putting the light particle in the center (0,0), webring to number of invariants up to five. Further increasing of quantity of discreteimpulses do not lead to new invariants, i.e., we need construct linear independent vec-tor of collision for every new particle. For light particles, it is possible to make in thefollowing way:

We add symmetrically four particles with impulses p5,6 = (1,∓1) and p7,8 =

(−1,±1) and proceed colliding them with central particle: e0514, e06

12, e0723, e08

34. Here wedo not have new invariants. Then we introduce the particle with impulses p9 = (2,0)and collision e09

56 along with axial and diagonal symmetric analogs and so on, fillngup all the surface by light particles. In order that do the same with heavy particles,we should put the particle (0, 0) in center, thus increasing the number of invariants upto six. Adding the heavy particles consequently just like we did with the light ones,

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we will obtain one extra invariant. To eliminate it we need to introduce at least onemore collision between light and heavy particles, which vector is linearly independentfrom previously introduced. Such collision desined with respect of (12.1.3) will fixthe relation of masses M/m. Therefore, we proved a proposition that our model do notpossess any extra invariants.

12.4 On Solution of Diophantine Equations of ConservationLaws and Classification of Collisions

How do we solve the system of equations (12.1.3)? A rather good idea is to introducea new vector a for the first of equations (12.1.3), such that

p′ = q+ a,

p= q′+ a.(12.4.1)

Substituting (12.4.1) into the second equation of (12.1.3), we obtain(q′2− q2

)(M

m− 1

)= 2(a,q′− q). (12.4.2)

Assuming q′2− q26= 0 we get

M

m= 1+ 2

(a,q′− q)

q′2− q2. (12.4.3)

The relation (12.4.3) reveals the possible relations for masses of light particles withgiven impulses q and q′.

An earlier assumption q′2 − q26= 0 corresponds to collisions without interchange

of energy: from (12.4.1) it follows that p2− p′2 = 0, thus an amount of energy does

not change.Also we should note that (a,q′− q)= 0 means either same masses or the case

without exchange of energy (12.1.3). In this case parallelogram qq′pp′ becomes arectangle. Therefore, collisions of the heavy and light particles one can classify inthe following way:

1. collisions between heavy particles;2. collisions between light particles;3. collisions between different particles without interchange of energy;4. collisions between different particles with interchange of energy.

In one-dimensional case, (12.4.2) can be divided by q′− q 6= 0 (because forq′− q= 0 collision is absent):

(q′+ q)

(M

m− 1

)= 2a. (12.4.4)

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12.5 Boltzmann Equation for the Mixture inOne-Dimensional Case

In one-dimensional case, sphere is replaced by two points and collision law (for heavyparticle) becomes

p′ = (p+ q)M

m+M±

mM

m+M

( p

M−

q

m

).

Upper (positive) sign leads to identical transformation, and lower (negative) givesnontrivial unknown transformation. We have

p′ = (p+ q)M

m+M−

mM

m+Mu,

q′ = (p+ q)M

m+M+

mM

m+Mu, (12.5.1)

u=( p

M−

q

m

).

If m=M, then p′ = q and q′ = p and exchange of impulses occures. Hence, col-lision integral J( f , f ) for the particles of the same kind is equal to zero and for one-dimensional case, we have Boltzmann equation without such integral:

∂F

∂t= J( f ,F)=

∫ [F(p′)f (q′)−F(p)f (q)

]σ(u)|u|dq,

∂f

∂t= J(F, f ),

where p′ and q′ are defined by formulas (12.5.1).

12.6 Models in One-Dimensional Case

Considering one-dimensional case for collisions we use formula (12.4.4). Let Mm = 3.

Then from (12.4.4) gives a= q+ q′. A simplest nontrivial solution of that equationwith a= 1, q= 0, q′ = 1 gives p= 2, p′ = 1. As result of symmetrization, we obtainthe following model, represented by Figure 12.1:

Model I (see [291],[293]): Mm = 3, 3 light and 4 heavy particles.

0

Figure 12.1 Model I:M

m= 3, 3 light and 4 heavy particles.

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Impulses of light particles are denoted by dark dots, heavy ones—by big circles.Light particles posses impulses—0,±1, and heavy ones—±1,±2. This model admits

three reactions with cross-sections σ 0211

, σ 0(−2)(−1)(−1)

, σ 1(−1)(−1)(1)

different from zero. The

impulses of particles are enumerated, and impulses of heavy particles are denotedby tilde.

Now we can check whether a given model posesse a correct number of invari-ants: r+ d+ 1= 1+ 2+ 1= 4. Using induction process, we start from particles

(1, 1,−1,−1) and reactions between them σ−11−11

. Thus we have 4− 1= 3 invariants.

Adding particles 0 and 2 and reaction σ 1102

, one obtain 3+ 2− 1= 4 invariants. Adding

particle (−2) and remaining reaction, one get the same number of invariants—4 andthe corresponding system of equations. The light particles are described by functionsfi, i= 0,±1, and heavy ones by Fi, i=±1,±2. Then

∂f0∂t= σ 02

11( f1F1− f0F2)+ σ

0−2−1−1

( f−1F−1− f0F−2).

The above expression reveals that function f0 is involved in two reactions. Also wehave,

∂f1∂t+

1

m

∂f1∂x= σ−11−11

( f−1F1− f1F−1)+ σ1102( f0F2− f1F1).

A similar equation also holds for f−1.The equations for heavy particles can be derived in similar manner: F1 has a veloc-

ity 1M =

13m and is involved in two reactions; F2 has velocity 2

M =2

3m and is involvedin one reaction.

The model presented here is a unique one-dimensional symmetric normal modelconstructed until now.

12.7 The Models in Two-Dimensional Cases

The first example of the two-dimensional model in Figure 12.2a can be obtained bysymmetrization of the one-dimensional model:

Model II (see [291],[293]): Mm = 3, 5 light and 8 heavy particles, with a total number

of equations 5+ 8= 13.

Exercise 12.2. Write collection of collisions, prove that model is normal and write thecorresponding system of equations.

In two-dimensional case (in contrast to the one-dimensional) one can constructmodel with arbitrary relation of masses using induction process, both symmetric andnormal one. If we would like construct model with a small number of equations, thenwe should take one of the solutions of Diophantine equation (12.4.2) with energyexchange. Symmetrize it, then add possible collisions and prove that an obtained setof impulses and reactions corresponds to normal model.

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0

0

(a) (b) (c)

0

q' p'

p

q

Figure 12.2 (a) Model II; (b) Model III; (c) Model IV.

Model III (see [291],[293]): Mm = 7, 6 light and 6 heavy particles. Model is

symmetric with respect to axes, not diagonals Figure 12.2c. The basic reaction withexchange of energy is represented in Figure 12.2b, where

q= (0,−1), q′ = (−1,1), p= (0,3), p′ = (1,1).

By symmetrization with respect to axes, one obtain the following picture, presentedby Figure 12.2c.

Exercise. Write all reaction, prove that model is normal and write equations.

Model IV: Mm = 7, 8 light and 8 heavy particles. It was obtained from the third

model by symmetrization with respect to diagonals.Additionally, we would like to discuss a multiple-dimensional case, generalizing

Model I and Model 2 with Mm = 3. We can construct a model with 6d+ 1 particles.

Here 2d+ 1 are the light ones with impulses q0, . . . ,q2d and 4d—heavy ones withimpulses p1, . . . ,p4d, where d is a dimension of physical space. The components ofimpulses for light particles are:

p0,i = 0, p2k−1,i = pδi,k, p2k,i =−pδi,k, i,k = 1, . . . ,d.

The components of impulses of heavy particles are given below:

p4k−3,i = pδi,k, p4k−2,i =−pδi,k, p4k−1,i = 2pδi,k, p4k,i =−2pδi,k.

For them the following linear independent collisions are defined:

d− 1 collisions of the form e2k−1,2k+12k,2k+2 between light particles;

2d− 1 similar collisions between heavy particles;

2d+ 1 collisions between heavy and light particles e1,22,1

, e4k−3,2k−14k−1,0

and e4k−2,2k4k,0

.

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Number of invariants for this model is equal to 6d+ 1− (5d− 2)= d+ 3. Also wewant to outline that in one-dimensional case d = 1 the described model is consideredto be unique normal (up to present time). There exist a hypothesis that another sym-metric one-dimensional normal models do not exist.

Papers [69], [79], [290], [291], [293], [294] have some other examples of symmet-ric DM-BEs in R2 for different relation of masses equal to 2, 5, 7, etc. We have thefollowing construction procedure for discrete models of mixtures with a small numberof velocities:

1. Take one (basic) reaction with energy exchange between components.2. Making it symmetric with respect to coordinate axes, obtaining semisymmetrical model.3. Making it symmetric with respect to diagonals, which provides a symmetrical model.4. Check the quantity of invariants and choose models with correct numbers.

In two-dimensional case an introduced above model with thirteen impulses is min-imal symmetrical model. Among semi-symmetrical (i.e., symmetrical with respect toaxes only) models the minimal one contain nine velocities [79]. Further classificationof two-dimensional models and construction more complicated ones could be foundin [69], [79], [294].

12.8 Conclusions

In order to construct discrete models for mixtures with correct number of invari-ants, we need to solve the system of Diophantine equations for conservation lawsof impulse and energy (12.1.3). Just one solution with energy exchange makes pos-sible a construction of discrete model by the following scheme: (a) symmetrization;(b) calculation of invariants; (c) adding particles and reactions in the case of necessityto eliminated extra invariants; and (d) writing down the final system of equations.

12.9 Photo-, Electro-, Magneto-, and Thermophoresisand Reactive Forces

The term photophoresis was proposed by Felix Ehrenhaft [100]. In his experimentsdust, silver, and copper particles in gases irradiated by light “strongly exhibiteda tremendous lightnegative movement, although they ought to be most heated onthe side toward the light, and would expect a movement away from the light”(see [187]).

Movement toward light was called lightpositive or positive photophoresis and awayfrom it lightnegative or negative photophoresis. “During the course of the experiment,the motion of the particle traced out a spiral path. However upon magnification of agiven section of a given spiral, one saw a spiral path within the path of the largerspiral. . . . In viewing these microphotographs, one had the distinct impression thatsomething phenomenal was happening, but no definitive explanation for the obser-vation was presented” [187].

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Theory of photophoresis usually was considered in the framework of P.N. Lebedevirradiation pressure and heat effects. For a review, see [70]. “A unilateral aerosol parti-cle is affected by a photophoretic force and a radiation pressure force. The former is ofa radiometric nature and is a result of gas molecules interacting with the non-uniformlyheated particle surface” (this is the beginning of the paper [70]). “Depending upon thesize and optical properties of material of a particle both irradiated and shadowed sideof a particle can become more heated. That is why both negative and positive photo-phoresis could take place” [313]–[315].

Our explanation is based on reactive forces. A particle evaporates molecules on irra-diated side—this is the cause of positive photophoresis. This explanation was acceptedin comet astronomy from the 1950s, when the American astrophysicist F. Wipple sug-gested reactive forces connected with sublimation of comets. In laser thermonuclearsynthesis reactive forces are also well known (they call it ablation sometimes). Butmathematical treatment of this kind of process goes back to Maxwell [193]. The com-bination of those ideas was considered in [298, 300]. Paper [312] is to some extent adirect experimental justification of this topic.

Negative photophoresis is considered on the basis of counterreactive (or antireac-tive) forces. The light makes a surface to adhere molecules: the force of momentum ofadhered molecules is less (maximum twice) than of elastically reflected ones. That iswhy additional pressure toward the source of light appears [298, 300]. This is a simpleexplanation of negative photophoresis.

This movement has to be considered in the framework of more general class ofmovements—chemoreative ones. It was discovered that big particles moves at anyphysical-chemical process [196]—the motion was called chemoreative. Mathematicalmodel of that motion was created in [20, 197, 296]. This model takes into accountnot only forces but torques as well: that is why together with Newton second law weconsidered Euler equations for solids. Exact solution of the equation showed spiraltrajectories [296]. In [296] we constructed the model and got spiral paths for simplestcase of spherical particles, and in [20, 21] more general case of any convex particlewas studied.

So we got a simple explanation for negative photophoresis and Ehrenhaft spiralpaths independently (i.e., being out of knowledge of Ehrenhaft) and in more gen-eral situation. This gives explanation of mysterious Ehrenhaft helixes [100, 187].Inversely, Ehrenhaft spirals justify our general mathematical model [20, 197, 296]experimentally.

Ehrenhaft himself carried out experiments on electrophoresis and magnetophoresisas well [187] and also obtained helixes. He was an outstanding and recognized exper-imental physicist. He was a director of Institute of Physics in Vienna, and in 1938 hecame to the United States. But his theoretical attempts to explain his helixes were criti-cized. For instance he tried to explain helixes in electrophoresis by introducing chargessmaller than electrons. His kind of thinking is understandable as the only helixes onecould extract from physic textbooks were Lorenz forced ones. But the radiuses andsteps of spirals were different as we shall see from our exact solution. This explana-tion met strong opposition of Robert Andrews Milliken, a Noble prize winner for his

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crucial experiment on a minimal charge. Spirals in magnetophoresis Felix Ehrenhaftried to explain in similar manner (Lorenz forces) by magnetic monopoles.

Magnetic monopoles were introduced by Pole Dirac in analogy with electrons inorder to make Maxwell electromagnetism equations symmetric with respect to elec-tric and magnetic fields. Ehrenhaft theoretical conclusions were declined (but notexperiments) by Dirac, Einstein, and others [187]. Their objection was—magneticmonopoles or smaller than electron charges can be obtained experimentally (if theyexist) only for much higher energies.

So the experiments were recognized but not explained. We see now that Ehrenhaftexamined if not so fundamental phenomena, but more close to reality, more often andmuch more useful for technical applications: it was an example of a spiral path of abig (1/100—1/10000000 meters) particle in any physical or chemical process.

In [187] we have the following eloquent passage: Also curious is the fact that thewinding shapes of some of this spirals in the microphotographs reminded me of theshapes described by Nicola Tesla with respect to Plate 48 in which Tesla wrote: “Oneof the streamers is wonderfully interesting on account of the curiously twisted andcurved appearance. It is hard to conceive how a discharge can pass through the air inthis way when there exists a strong tendency to make it take the shortest route”. Thekeyword here is discharges as there is no any model for twisted discharges. It seemsto be also the spirals from [296] in this case of northern lights.

12.9.1 Model and a System of Equations

Here we follow [296]. A model can be described as follows. A particle is considered asa ball, which is determined by coordinates of center of mass R, momentum Q, angularmomentum K, and the unit vector S directed from a center of mass to the center ofactive zone (Fig. 12.3).

The forces that act on a particle are determined by collisions and a frequency ofcollisions has the usual in Kinetic theory expression:

d�= σ · (u,n) · θ((u,n)) · f (r,p) · dr · dp.

Here σ = σ(R,Q,r,p)—is a crossection of collisions {R,Q,S,K}+ {r,p} → . . . thatfor rigid spheres have the form

σ = σ(R,Q,r,p)= δ(|r−R| − ρ), θ(x)=

{1, x> 0,0, x≤ 0

.

δ(x) is a Dirac δ—function, u=p

m−

Q

Mis a relative velocity of a molecule and a

ball, ρ is a radius of a particle, n—is a unit vector orthogonal to the surface at thepoint r = R− ρn (see Fig. 12.4), f (r,p) is a distribution function of molecules overmomentum p and space r.

We assume that some molecules are elastically reflected from the surface of theparticle, and some of them adhere (the adhesion here models chemical interaction).

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Q

p

S

K

Q

Rr

Figure 12.3 A molecule has a momentum p and space coordinate r.

u

n

dσ

Figure 12.4 A picture of a collision.

We assume that the fraction of adhering particles at the point of the ball with the

internal normal n is β(n), and we put A=∫

S2

n ·β(n)dn and orientation vector S=

A

|A|. Dynamics of this vector is described by the equation

dS

dt=

1

J· [K,S], where J is

a moment of inertia.A system of equations can be written in the form

dR

dt=

Q

M,

dQ

dt= Fchem+Felast,

dK

dt=Mchem,

dS

dt=

1

J[K,S].

(12.9.1)

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The first equation is the definition of a velocity; the second is a Newton law withtwo terms. The first term is connected with inelastic collisions and has the form

Fchem = ρ2∫∫

R3×S2

β(n,S){p− 2µ(u,n)n}(u,n)θ((u,n))f (R− ρn,p)dndp. (12.9.2)

The second term is determined by elastic collisions

Felast = 2µρ2∫∫

R3×S2

n(u,n)2θ((u,n))f (R− ρn,p)dndp. (12.9.3)

The third equation describes changing of a moment dK/dt and connected only withinelastic collisions (in the case of ball):

Mchem = µρ3∫∫

R3×S2

β(n,S)[u,n](u,n)θ((u,n))f (R− ρn,p)dndp. (12.9.4)

Here µ=mM

M+m—is the reduced mass, ρ is the radius of the ball particle, u is the

relative velocity.

12.9.2 System of Equations in Maxwell Equilibrium for SmallVelocities of a Particle

We take destribution function as Maxwellian

f (t,r,p)= n0(2πmkT)−32 · e−|p|2

2mkT .

Assume that the ratio of the particle velocity and the mean square velocity of gasmolecules is small. Then we get the following expressions for integrals (12.9.2)–(12.9.4):

Fcomp =−8√

2π · ρ2n0kT

3 · (1+ ε)· ν,

Fchem =2ρ2n0kT

1+ ε·

{−

1− ε

4

∫β(n)ndn+

3− ε

2√π

∫β(n)(ν,n)ndn

},

Mchem =−ρ3n0kT

1+ ε

√2

π·

∫β(n)[ν,n]ndn.

Here ν =Q

M

√m

kt—is a ratio of velocity of a body

Q

Mand heat velocity, ε =

m

M—is a

ratio of masses of molecule and a particle.

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Let us consider β symmetric with respect to rotations around S. That means that βis a function of only one variable

β = β((n,S)). (12.9.5)

Then the system gets the following form:

dR

dt=

Q

M,

dQ

dt= (χ0+χ1 · (Q,S)) · S− λ ·Q,

dS

dt=

1

J[K,S],

dK

dt= γ [Q,S].

Here one has by integrating with Maxwellian and taking in mind (12.9.5):

χ0 =1− ε

1+ επρ2n0kT

1∫−1

β(ζ )ζdζ,

χ1 =ρ2n0 ·

√2πmkT

2M·

3− ε

1+ ε·

1∫−1

β(ζ )(3ζ 2− 1)dζ,

λ=ρ2n0 ·

√2πmkT

M(1+ ε)·

8

3−

3− ε

2·

1∫−1

β(ζ )(1− ζ 2)dζ

,γ =−

ρ3n0 ·√

2πmkT

M(1+ ε)

1∫−1

β(ζ )ζdζ.

Now we shall assume that the rotation is almost constant. Then we get the followingsystem

dR

dt=

Q

M,

dQ

dt= (χ0+χ1(Q,S))S− λQ, ω =

K

J≈ Const,

dS

dt= [ω,S].

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Integration of this system shows that the trajectory R(t) tends to cylindrical spiral witha constant step L and diameter D:

L=2π (P∞,ω)

Mω2, D=

2√

P2∞ω

2− (P∞,ω)2

Mω2. (12.9.6)

Here we have the dependence upon momentum P in the limit as time tends toinfinity. Those formulae (12.9.6) give steps and diameters of cones that Ehrenhaftcould obtain and it depends upon parameters in absolutely different from Lorenz man-ner. For more details see [20, 21, 197, 296, 299].

The case of general form of a particle was considered by Batisheva [20, 21].Positive and negative photophoresia are quantum effects and have to be explained

through photo effect [300]. In both cases photoeffect can be internal (without emissionof electrons but raising them on a higher energy level) or external: internal is perhapsmore appropriate for explanation of photophoresis. If photoeffect induces photo-sublimation (or photodisintegration) we have positive photophoresis. And if it givesphoto-adsorption one has negative photophoresis. It seems that even the same sub-stance can exhibit negative or positive photophoresis in dependence upon frequencyor intensity of light.

This quantum consideration [300] differs from [312] where positive photophoresiswas explained through Greenhouse effect and thermophoresis. It would be interestingto create more detailed model to explain Ehrenhafts “spirals within the parts of otherspirals” [187] and to explain electrophoresis and magnetophoresis. And it has to besupported by experiments—at least to repeat Ehrenhaft’s ones. But to disentangle thisis beyond the scope of this article. Let say a few words on other phenomena.

1. Electrophoresis. This is a movement of a big particle in gas or in liquid in electric fields.Sometimes they call it also photophoresis (a movement in electromagnetic field). Usuallythey explain it as a movement of a charged particle, but the process of ionization is difficultto explain for weak electric fields. But the former quantum mechanic view and the samemathematical model works. Again Ehrenhaft spirals give experimental justification.

2. Magnetophoresis. This is a movement in magnetic field. The model is the same. It is inter-esting, that those two terms seems to be introduced by Ehrenhaft also, and he wrote elec-trophotophoresis and magnetophotophoresis [101].

3. Thermophoresis. This is a movement of a big particle in gas or liquid under temperaturegradient. Perhaps sometimes reactive forces are important in this case also. For instant, heat-ing may induce adhesion. In other way it is difficult to clarify negative Thermophoresis (if itexists) [19]. And to wait for helixes in experiment.

In all those cases more detailed models have to wait for experiments.

12.9.3 Applications

Chemoreative motion is widely spread as we have it for any big particle, chemicallyor physically reacted with gas. So it must have a tremendous number of applications.Direct and indirect photophoresia have several applications.

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It was already discussed for comet astronomy and for possible explanation of formsof Northern Light. A number of applications are considered for planet astronomy inpaper of G. Wurm and O. Krauss [312]. There we also read “One might think ofcreating an artificial surface with much stronger, optimized photophoretic forces. Inanalogy to solar sails based on radiation pressure solar sails based on photophoresiscould be much stronger. These could, e.g., be used for propulsion of small probes onMars or in Earth’s stratosphere.” The idea was independently discussed in [300] in thefollowing words:

“Photo-reactive engine. On the basis of direct photophoresis a scheme of photo-reactive engine can be proposed. A layer of substance, sublimating under photons,covers a surface of a rocket or some part of it appearing in proper moment. Now theytry to construct solar sails and to use irradiation pressure. Reactive forces can help,as they are much stronger. If to use not only photons but other components of solarwind such an engine can be called Solar-reactive.”

It is interesting for applications as well that in [312] and [300] (and here we fol-low [193, 298, 300]) explanations of direct photophoresis are different. G. Wurm andO. Krauss use thermophoresis and solid state Greenhouse effect. Photoeffect mech-anism gives another picture from quantum mechanical point of view. Thermoeffectsare connected with rotation and vibration degrees of freedom and photoeffect does notenlarge temperature. That is why it will be better to use photoeffect in those enginesand perhaps even to fight against thermophoresis. So both views from [300] and [312]give complementary pictures as they work for different intensities and/or frequencies.Another application that was discussed in [300] is acceleration of particles by laserbeams as it in usage for laser thermonuclear synthesis.

Both indirect and direct photophoresia can be applied for dusty plasma, forNorthern lights. Now it is clear that Felix Ehrenhaft spiral paths and so both positiveand negative photophoresia have their explanation in the framework of chemoreativemotion: former as a consequence of reactive forces and the latter of counterreactiveones. By contrast, Ehrenhaft spiral paths strongly support all mathematical theoryof motion of any big particle in reacting gas (chemoreative motion), constructed inpapers [20, 196, and 296]. It is interesting to explain spiral path within the path of thelarger spiral—this perhaps could be done by more detailed model.

12.9.4 Conclusions

1. Mathematical theory is constructed for a movement of a big particle interacted physically orchemically with gas. Especially positive and negative photophoresia, electrophoresia, mag-netophoresia, and thermophoresia got some explanation.

2. Exact solutions are constructed for system of equations of rigid body motion. Are proposedideal helix trajectories as asymptotes for any solution as time tends to infinity. This part ofjob was in principle carried out by Batisheva [20, 21, 197, 296, 299].

3. Experiments of Felix Ehrenhaft were explained and spiral paths got their mathematicaljustification.

4. Several new experiments are proposed. Especially to repeat Ehrenhaft ones and to comparediameters and steps of cones with exact formulae (12.9.6).

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13 Quantum Hamiltonians andKinetic Equations

In this chapter we consider the correspondence between quantum Hamiltonians andclassical kinetic equations of the chemical kinetics type. Namely, it will be establishedon the basis of conservation laws. Both quantum Hamiltonians describing the pro-cesses of creation and annihilation of particles and kinetic equations modeling similarclassical systems use conservation laws.

Conservation laws linear by particles densities play important role in the theoryof kinetic equations. In the case of Boltzmann equation they are fundamental macro-scopic values necessary for introduction of continuous medium, when the hydrody-namics equations for mean values of density, impulse and energy are written (seeChapters 9, 10, and 12 of this book; [116], [173]). In the homogenius space case ofthe Boltzmann equation, these laws completely define the qualitative behavior of thesystem. H- theorem justify tending of the system to stationary distribution, whichparameters are defined from the corresponding conservation laws. For the classicalBoltzmann equation, this distribution is called the Maxwellian.

The idea of this correspondence (Landau-Lifshits-Sewell-Streater [177]) allows towrite down the generalizations of discrete models for Boltzmann equations based onconservation laws for quantum Hamiltonians (QH) and kinetic equations (KE). Eitherfor cases of annihilation and particle creation or triple (and even higher) order colli-sions. This generalization refers to a class of equations in chemical kinetics where theH- theorem holds.

In this chapter, we define the space of linear conservation laws for polynomialquantum Hamiltonians when an operator depends on a number of particles; we alsoconsider their classical analogs. Second, we consider similar conservation laws forkinetic equations, revealing the correspondence between QH ↔ KE and provingH- theorem for them. We study conservation laws for quantum and classical casesto describe the process of Raman scattering throughout the study of spectrum of poly-nomial Hamiltonian.

13.1 Conservation Laws for Polynomial Hamiltonians

Let us consider tensor product F = Fp (Fock space) of p Gilbert spaces with the basisen1⊗en2 · · · ⊗ enp , where eni , ni = 0,1, . . .—a usual unit vector (for the details of tensor


http://dx.doi.org/10.1016/B978-0-12-387779-6.00013-2

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multiplication, see [38] or Chapter 11). Also we define conjugate operators a−k , a+kacting over F by the rule

a−k |n1〉|n2〉 · · · |np〉 =√

nk|n1〉|n2〉 · · · |nk− 1〉 · · · |np〉,

a+k |n1〉|n2〉 · · · |np〉 =√

nk+ 1|n1〉|n2〉 · · · |nk+ 1〉 · · · |np〉.(13.1.1)

Here operators a−k , a+k do satisfy the standard bozon rules of commutation:[a−j ,a

+

k

]= δjk. (13.1.2)

These operators usually are called the operators of annihilation and creation, whileoperator a+k a−k ≡ nk is named as an operator of number of particles, since its eigen-values are integer numbers nk-s, and eigenvectors represent the already mentionedstandard basis.

Many problems of quantum optics are described by Hamiltonians which are poly-nomial by operators of creation and annihilation. For example, the classical model ofRaman scattering for the one-dimensional radiation on excitations (phonons) of crystallattice reads as

H =2∑

k=0

ωka+k a−k +B(a+0 a−1 a−2 + a+1 a+2 a−0

),

where ωk and B are real constantes. The generalizations of this model are studied inSection 13.3. Even for the simplest nonlinear model of this kind, we need to know theconservation laws to be able to study the spectrum of Hamiltonian and its asymptoticproperties.

Let us consider polynomial Hermitian operators of the general form

H = H0+∑

(α,β)∈J

bαβa+αa−β + h.c. (13.1.3)

Here H0 = H0(n1, . . . , np

)is a diagonal operator over the given basis:

H0|n1〉 · · · |np〉 = H0(n1, . . . ,np

)|n1〉 · · · |np〉,

bαβ—real valued coefficients; aα = aα11 · · ·a

αpp ; (α,β) ∈ J ⊂ Z2p

+—nonnegative integermulti-indices, i.e., the vectors from some subset J of integer 2p-dimensional latticeZ2p+ ; letters h.c. denote Hermitian conjunction. By J1 we denote the set of vectorsα−β ∈ Zp, and L—its linear hull.

What kind of conditions must be set upon the set J of values of multi-indices toguarantee that operator (13.1.3) possess conservation laws linear in number of parti-cles? The answer has been obtained in [292]:

I =p∑

k=1

µknk (13.1.4)

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Quantum Hamiltonians and Kinetic Equations 229

is an operator that commutes with Hamiltonian (13.1.3), if vector µ= µ1, . . . ,µp) isorthogonal to linear hull L of vectors α−β, i.e., belongs to the orthogonal complementL⊥ in Rp. Indeed, from equations (13.1.1), (13.1.2) follows:[

I, H]=

∑(α,β)∈J

∑k

µk(αk−βk)(bαβa+αa−β − h.c.

). (13.1.5)

Then for arbitrary vector µ ∈ L⊥, we obtain [I, H]= 0. Thus, Hamiltonian (13.1.3) hasp− d (d ≡ d (J1)= dimL) linear independent conservation laws of the form (13.1.4).

Which interpretation has this conservation laws in the case of classical Hamiltonfunction? We associate a function of 2p variables with operator (13.1.3):

H = H0

(|z|2

)+

∑(α,β)∈J

bαβ zαzβ + c.c., (13.1.6)

where c.c. denotes complex conjunction, and canonical variables z, z satisfy Hamiltonequation:

zk =−i∂H

∂ zk, ˙zk = i

∂H

∂zk.

Then expression

I =p∑

k=1

µk|zk|2 (13.1.7)

is the first integral of the system (13.1.6) taken for vectors µ, which for all non-trivial indices α,β from (13.1.6) satisfy condition (µ,α−β)= 0. Conservation laws(13.1.7) are well-known in classical mechanics (see [58], [212] for example). Theywere used to study the stability of the solutions and the study of integrability.

13.2 Conservation Laws for Kinetic Equations

Kinetic equations describe changes of distribution function of the particles in chemicalreactions, collisions, and so on. A simplest example of this equations are the balanceequations for density of particles in binary collisions, when transition of the particlesof one kind into another one is proportional to the density of colliding particles. Theso-called four-component Maxwell-Braodwell model serves as a perfect example:

dn1

dt= n3n4− n2n1 ≡ f1,

dn2

dt= f1,

dn3

dt=

dn4

dt=−f1.

The discrete models of Boltzmann equation (DMBE) are the generalizations of thismodel (see [116], [264], and Chapters 10 and 12 of this book):

dni

dt=

∑jkm

B ijkm(nknm− ninj), i= 1, . . . ,p. (13.2.1)

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Here n ∈ Rp+—vector of p-dimensional linear space with positive components, B ij

km =

Bkmij = B ji

km > 0—positive constants (cross-sections of collisions) for reaction of theform (i, j)→ (k,m).

Summation in (13.2.1) goes over all reactions in which participates i-th matter.H- theorem is satisfied for this system as well as for Boltzmann equation: functional

H =p∑

i=1ni lnni decreases due to system (13.2.1), i.e.

∂H

∂t≤ 0.

It is easy to check that∑µini is conserved for system (13.2.1) if for all nontrivial

cross-sections of collisions holds the equality µk+µm = µi+µj. With every reaction

with nontrivial cross-Section B ijkm we associate two vectors eij and ekm ∈ Zp. They

have ones in the places indicated by indices, and all other elements are zero. Hencean existence condition of linear conservation laws µk+µm = µi+µj is written as(µ,eij− ekm)= 0.

The mentioned expressions has the same form as conservation laws (13.1.4) and(13.1.7) representing quantum and classical systems. Therefore, one can establish cor-respondence between quantum Hamiltonians and kinetic equations based on existenceof the similar conservation laws.

Noting that quantum Hamiltonian modeling point wise collision of two particles(of the k and m kinds), providing as the result particles of i and j kinds (and vice versatoo) is written as

H = H0+∑ijkm

bijkma+k a+ma−j a−i + h.c.

One can obtain a Hamiltonian in the form (13.1.3) just by denoting α = eij, β = ekm.Moreover, as it follows from (13.1.5) operator (13.1.4) commutes with H when(µ,eij− ekm)= 0 for all indices (k,m), (i, j) where bij

km > 0.From here follows the natural comparison of kinetic equations of the form (13.2.1)

and quantum Hamiltonians (13.1.3), making possible the generalization of DMBE foran arbitrary power reactions. For each Hamiltonian of the form (13.1.3), we establishthe correspondence with the system of equations with the same conservation laws:

dnk

dt=

∑(α,β)∈J

(αk−βk)Bαβ (n

β− nα). (13.2.2)

It is easy to see that (13.2.2) has the same conservation laws as for (13.1.4):

I =p∑

k=1

µknk = Const, µ ∈ L⊥. (13.2.3)

However, system (13.2.2) becomes an exact kinetic analog of quantum evo-lution equations for operators, depending on the number of particles (or corre-sponding classical dynamical systems), because (13.2.2) has decreasing functionalH =

∑k(nk lnnk− nk), which complies with the H- theorem:

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dH

dt=

∑k

lnnk

∑(α,β)∈J

B αβ (αk−βk)(n

β− nα)=

=

∑(α,β)∈J

B αβ (n

β− nα)ln

(nα

nβ

)≤ 0. (13.2.4)

Now we are ready to extend the correspondense (13.1.3)↔(13.2.2) onto continuouscase.

1. Scatttering of particles with energy E(p) over potential

U(x)=1

2π

∫eipxV(p)dp

has a corresponding Hamiltonian

H =∫

E(p)n(p)dp+∫∫

V(p− p′)a+(p)a−(p′)dpdp′

and transport equation:

df (p)

dt=

∫σ(p,p′)

(f (p′)− f (p)

)dp′.

2. Four-particle Hamiltonian

H = H0+

+

∫dp1dp2dp3dp4

(w(1,2→ 3,4)a+(p4)a

+(p3)a−(p2)a

−(p1)+ h.c.)

has a corresponding equation of Boltzmann type:

df (p1)

dt=

∫dp2dp3dp4σ(p1, . . . ,p4)( f (p3) f (p4)− f (p2) f (p1)).

3. Hamiltonians of quantum electrodynamics are introduced in a convenient form in thebook [104]. Using the introduced technique we can introduce the kinetic equations relatedto them. But the problem for calculating kinetic coefficients Bαβ introduced by constantes ofinteraction bαβ is still actual: for example, finding velocities of chemical reactions in chem-ical kinetics is a fundamental problem. The relation with quantum electrodynamics is quiteuseful, since we can find an approximate solution from perturbation theory. By contrast,kinetic radiation transport equations are widely used nowadays, but calculation of cross-sections by formulas of perturbation theory is still under study.

4. Chemical kinetics. Equations like (13.2.2) are equations describing chemical reactions forwhich velocities of direct and inverse reactions coincide. However in reality, direct andinverse reactions oftenly have different velocities. Therefore it is interesting to consider themfrom point of view of H- theorem and conservation laws.

Quantum kinetic equations need further generalizations. In [252], [263] are intro-duced discrete models of quantum Boltzmann equation (Uehling-Uhlenbeck equa-tion). In [177] the analogy between kinetic equations and quantum Hamiltonians

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were used for description of properties of dialectrics. The general discrete modelsof Uehling-Uhlenbeck equation are introduced and studied in [199], [201], [287]. Inparticular cases mentioned in [242], such models appeared in modeling of chemicalkinetics.

Here we give necessary generalizations of the mentioned above results, being ableto derive an analog of the law (13.2.4).

Let H(n) is some function and hi = e∂H∂ni . Let B α

β (n)—the collection of positivefunctions. Consider the system of differential equations:

dni

dt=

∑(α,β)

(αi−βi)Bαβ (h

β−hα). (13.2.5)

This system also possesses conservation laws of the form∑µini, for (µ,β −α)= 0.

Functional H also decreases:

dH

dt=−

∑(α,β)

Bαβ (5H,α−β)(e(5H,α)− e(5H,β))≤ 0.

Therefore, one can construct the functional H over arbitrary collection of stationarypoints gaining its minimum in those points and obtain many examples of equationsfrom chemical kinetics with given attraction points. For example, assume N = 3 in(13.2.5). Taking the first matter—boson, second, and third—fermions. Second matterdenotes fermion in excited state. It emits photons (bosons), hence we obtain the thirdmatter—nonexcited fermion. Under this assumptions we obtain the system:

dn1

dt= B

(n2

1− n2−

n1

1+ n1

n3

1− n3

)≡ g;

dn2

dt=−

dn3

dt=−g; (13.2.6)

B≡ B010101 = b(1+ n1)(1− n2)(1− n3), b> 0.

Dynamical system (13.2.6) is considered in semicylinder P : n1 ≥ 0, 0≤ n2,

n3 ≤ 1. These restrictions are permanent, H- function is convex in P. An introducedsystem have two conservation laws: n1+ n2 = C1, n2+ n3 = C2. The second oneexpresses conservation for the number of fermions. Hamiltonian from Section 13.1corresponding to system (13.2.6), has the form

H = H0+B(a+f +1 f−2 + f −1 f +2 a−

),

where a± are related with boson, and f±1,2—with fermions. In the same manner, onecan construct a relaxation model (13.2.5) for each quantum Hamiltonian (13.1.3) forbosons or fermions with H- function of the form (see 9.4):

H =∑

j

Hθj( fj), Hθ ( f )= f ln f − θ(1+ θ f ) ln(1+ θ f ),

where θ =+1 for bosons, θ =−1 for fermions and θ = 0 for Boltzmann statistics.

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13.3 The Asymptotics of Spectrum for Hamiltoniansof Raman Scattering

In this section, we consider the problem on asymptotics of spectrum for Hamiltonianof the form (13.1.3), for the case when conservation laws (13.1.4) convert the probleminto finite-dimensional.

13.3.1 Stokes Scattering

In quantum optics to describe a Raman scattering of one-mode radiation of frequencyω0 on excitations (phonons) of various kinds arising in medium, the following model(see for example [228]) is commonly used:

H =p∑

k=0

ωka+k a−k +B

(a+0

p∏k=1

a−k + h.c.

), (13.3.1)

where ωk, B are real valued constantes. According to (13.1.5), this system has p con-servation laws, written in the form

I0 = n0+ np, Ik = nk− np, k = 1, . . .p− 1. (13.3.2)

We will seek eigenvector of Hamiltonian decomposing it over standard tensor mul-tiplication basis of one-particle Fock spaces:

|ψ〉{3} =∑

j0 ... jp∈3

λj0... jp |j0〉 · · · | jp〉. (13.3.3)

Since operators (13.3.2) commute with Hamiltonian, the space of eigenvalues ofthese operators is invariant to Hamiltonian acting (13.3.1). Hence applying Hamilto-nian to every term of (13.3.3), values j0+ jp and jk− jp do not change. For simplic-ity we choose these constants in (13.3.3) to obtain j0+ jp = N, j1 = j2 = . . .= jp = j.Conservation laws provide establish bounds to the range of indices jk, therefore 3in (13.3.3) is a finite set of admissible values of completing numbers for the givenconstants in conservation laws (13.3.2). This choice uniquely defines the set 3 by thevalue of the constant N, and equation (13.3.3) becomes

|ψ〉N =

N∑j=0

λj|N− j〉0| j〉1 · · · | j〉p. (13.3.4)

Then the eigenvalues problem for Hamiltonian (13.3.1) over this subspace is writ-ten in the form of equation to determine value λj: H|ψ〉N = E(N)|ψ〉N or√

N− j(j+ 1)p/2λj+1+√

N− j+ 1(j)p/2λj−1 =

=

[E(N)−ω0N

B+

j

B

(−ω0+

p∑k=1

ωk

)]λj. (13.3.5)

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System (13.3.5) written with respect to j= 0. N also can be presented in matrixform

Dλ= xλ, x≡ x(N) =E(N)−ω0N

B, (13.3.6)

where D—symmetric three-diagonal matrix of dimension (N+ 1)× (N+ 1) matrixwith elements (no summing by j):

Djk =√

qj−1δj,k+1+√

qjδj,k−1+ cjδjk, qk = kp(N− k+ 1),

ck =k

B

(ω0−

p∑i=1

ωi

). (13.3.7)

Our first problem is study the behavior of eigenvalues of matrix Djk while N→+∞. This study [228] shows the domain of applicability of such models in quantumoptics. Let x(N)j eigenvalues of matrix Djk for any fixed N. To be able to detect major

asymptotics in N, we introduce matrix D with elements DjkN−(p+1)/2. Introducing thecorresponding eigenvalues

s(N) = x(N)N−(p+1)/2, (13.3.8)

and consider the traces µn of the nth degree of matrix D:

µn =1

N+ 1

N∑j=0

(x(N)j

Np+1

)n

=1

N+ 1tr(Djk)n. (13.3.9)

Tending N→+∞ and setting Q(z)= zp(1− z), from (13.3.7) we obtain (detailedcalculations are provided in [220]),

µ2n =

(2n

n

) 1∫0

Q(z)ndz+O

(1

N

)=

=

(2n

n

)(pn)!n!

[(p+ 1)n+ 1]!+O

(1

N

), (13.3.10)

µ2n+1 = O

(1

N

).

Hence, when N→+∞ momentums of properly normalized eigenvalues (13.3.8) aredefined and are finite. Then according to the theorem of spectrum radius

Ri = sups(N) = limn→∞

µ1/nn = 2

√pp

(p+ 1)p+1. (13.3.11)

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According to Xelli theorem [103] a distribution function for zeros s(N)k

ρN(s)=1

N+ 1

N∑k=0

δ(s− s(N)k ) (13.3.12)

has a limit point ρ(s), and as it follows from (13.3.10) momentums of the functionρN(s) have a limit at N→+∞

σ (N)n =

+∞∫−∞

ρN(s)snds,µn = lim

N→∞σ (N)n =

+∞∫−∞

ρ(s)snds. (13.3.13)

Relation (13.3.11) shows that Kalerman’s theorem [103] about one-to-one recon-struction of distribution function from its momentums is suitable in our case, too.Therefore, there exists a weak limit ρ(s)= lim

N→∞ρN(s). Applying Kalerman’s theo-

rem to the odd part of ρ(s), one obtain that odd part of ρ(s) vanishes due to (13.3.10).Therefore, the measure ρ(s) is symmetric with respect to zero. Returning now to amatrix presentation D, we obtain from (13.3.8) the existence of such eigenvalues thatx(N) ≈±BRN(p+1)/2.

Then from (13.3.6) follows that for p> 1 spectrum of Hamiltonian (13.3.1) is notseparated from minus infinity for any sign of constant B. This result is important,because it restricts the domain of validity for polynomial models in quantum optics,describing the interaction of radiation with matter.

13.3.2 Raman Scattering Considering Anti-Stokes Component

As discussed earlier, p-dimensional system (13.3.1) have p− 1 additional conservationlaws (13.3.2). However, even regarding a lower number of conservation laws, thesystem allows a similar aproach to study spectrum asymptotics. The key factor hereis fixing the signs of constantes for conservation laws. Doing so, we get the finite-dimensional system of equations on eigenfunctions. For example, for Hamiltoniangeneralizing, the model of so-called Stokes and anti-Stokes scattering [228]:

H =p+1∑k=0

ωka+k a−k +BSa+0

p∏k=1

a−k +BASa+p+1

p−1∏k=0

a−k + h.c.

This Hamiltonian describes interaction of p+ 2 quasiparticles, and possesses p con-servation laws:

nk+ np+1− np = Jk, k = 1, . . . ,p− 1; n0+ np+ np+1 = J0. (13.3.14)

By analogy with “Stokes”-only approach we introduce normalized eigenvaluess(N)k = x(N)k N−(p+1)/2 and consider asymptotics of momentum

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µn =1

r(N)

r(N)∑k=1

(s(N)k

)n

for N→+∞, where r(N)—the number of integer points on polygon defined by fixingconservation laws. We denote these constants eigenvalues of operators in (13.1.4), asNk and N, respectively. We’ll study asymptotics of momentum assuming

N→+∞, Ni→+∞,Ni

N→ Ai, i= 1, . . . ,p− 1.

Using the same techniques as before, we obtain

µ2n+1 = O

(1

N

),

µ2n =

(2n

n

) n∑k=0

(n

k

)2

B2kS B2n−2k

AS C2n+O

(1

N

)(13.3.15)

C2n =

∫∫x+ y≤ 1y− x≤ Ai

xkyn−k(1− x− y)np−1∏i=1

(Ai+ x− y)dxdy.

Since the number of conservation laws (13.3.14) is lower by 2 than a dimension ofthe system, we obtained in (13.3.15) a double integral instead of single-dimensionalin (13.3.10). And now once again Xelli and Calerman’s theorems give the existenceof limit measure and it finiteness for normalized eigenvalues.

13.4 The Systems of Special Polynomials in the Problemsof Quantum Optics

In this section, we study the system of eigenfunctions for the spectrum problem con-sidered in Section 13.3 for Hamiltonian (13.3.1). A similar calculations can be donein more general case.

According to Section 13.3, eigenvector of Hamiltonian (13.3.1) has the form(13.3.4). To determine its components λk, k = 1, . . . ,r(N) we use system (13.3.5).It can be simplified by introducing Pk(x) defined by formulas

Pk(x)≡ P(N)k (x)= ukλk(x), uk+1 = uk√

qk+1 (13.4.1)

where qk = kp(N− k+ 1) according to (13.3.7). Then (13.4.1) and (13.3.5) introducethe following system for Pk(x) (upper index is omitted to simplify the expressions):

Pk+1(x)= (x− ck)Pk(x)− qkPk−1(x), P0 = 0, P1 = 1. (13.4.2)

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Since (13.4.2) represents decomposition formulas for (k+ 1)-th minor by rowand denoting Mk(x) major minor of k-order for matrix |D− xI| (13.3.6), we obtainP(N)k+1(x)= (−1)kMk(x). Therefore, the problem of finding the eigenfunctions of spec-trum problem is reduced to recurrent relations (13.4.2), also known from the theory oforthogonal polynomials [269].

A major difference between (13.4.2) and classical polynomials involves numbersqk, which can be zero and negative values. Calculations are completed at step N+1, because qN+1 = 0. Then eigenvalues are the roots of polynomial P(N)N+1(x), equalto the matrix determinant in (13.3.6). For example, for p= 1 (the case of quadraticHamiltonian) the roots of polynomials are integer numbers:

P(2N−1)2N (x)= x

N∏k=1

(x2− 4k2), P(2N−2)

2N−1 (x)=N∏

k=1

(x2− (2k− 1)2).

13.5 Representation of General Commutation Relations

Section 13.1 studied conservation laws for quantum Hamiltonians in the case of classi-cal boson commutation relations (13.1.2). It was shown that a quantity of conservationlaws, which are linear by operator depending on the number of particles, is defined bydimension of linear hull of vectors α−β, composed from degrees of polynomials.How do these results depend on the form of commutation relations between conjugateoperators of creation and annihilation? How does the spectrum asymptotics depend onthem?

First, we consider which representations of generalizations of standard commu-tation relation can appear. The questions of representation theory of noncanoni-cal commutation relations has been investigated in many papers (see, for example,[202], [280], [281], [282], [284], [288]). Let us consider commutation relations of theform

a−a+ = f (a+a−). (13.5.1)

We construct representations of relations (13.5.1) with a− and a+—mutually con-jugate in some Hilbert space and (13.5.1) are satisfied on the basis of eigenvectorsof operators a+a− and a−a+. The corresponding eigenvalues are nonnegative. So, letsome real function f is defined on nonnegative semiaxes. Depending on its behav-ior, we differ between five different types of representations for relations (13.5.1).Denote as {en} some orthogonal basis of linear space (finite-dimensional or infinite-dimensional) consisting from eigenvectors of operator a+a−. Actions of operators a±

over the basis, we define in the following manner:

a−en =√λnen−1 a+en =

√λn+1en+1, (13.5.2)

where collection of indices n depends from representation.

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Substituting (13.5.2) into (13.5.1), we obtain recurrent relation defining eigen-values λn:

λn+1 = f (λn). (13.5.3)

Depending on behavior of sequence (13.5.3), we obtain the following representa-tions of relations (13.5.1).

1. Representation of boson type. It is given by formulas (13.5.2), (13.5.3) for n≥ 0, if λ0 = 0and function f guarantees that λn > 0 for all n> 0.

2. Representation of antiboson type. Also it is given by formulas (13.5.2), (13.5.2), but forn≤ 0. Moreover, λ0 = 0 and function f guarantees that λn > 0 for all n< 0.

3. Two-sided representation is given by formulas (13.5.2), (13.5.3) for all integer n, if functionf gives λn > 0 for all integer n.

4. Representation of fermion type. This representation is constructed in Rm+1 by the sameformulas (13.5.2), (13.5.3) taking 0≤ n≤ m. Here m is a positive integer, such that conditionλ0 = 0; λn > 0, 1≤ n≤ m; λm+1 = 0 holds.

5. Periodical representation appears if there exists such m, that λ0 = λm+1 > 0. Then weconstruct representation in finite-dimensional space Rm+1, having complemented formulas(13.5.2), (13.5.3) by conditions

a+em =√λm+1e0, a−e0 =

√λ0em. (13.5.4)

The methods of investigation of the spectrum for polynomial Hamiltonians givenin Section 13.3 remains valid for nonstandard commutation relations, when the parti-cles from different Hilbert spaces commute between each other (so-called generalizedPaulions). Examples of such study are given in [202], [288], [289], where the separa-bility criterion of the spectrum from minus infinity also was obtained.

13.6 Tower of Mathematical Physics

We obtain the following construction scheme for the structure of mathematicalphysics:

Quantum HamiltoniansLagrangian

Continuous medium

Boltzmann-type Kinetic equationVlasov-type Kinetic equation

Starting from Lorentz lagrangian (Vlasov-Maxwell; see Chapter 3), we are ableto construct nearly all physical phenomena. Nearly all refer to the existence of twoanother “physics”:

l micro-world, described by Vlasov-Yang-Mills lagrangian and its generalizations;l macro-world, described by Vlasov-Einstein lagrangian.

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To obtain “nearly all physics” we have to get sparse gases, liquids, and hard bodies,as well as of plasma state of matter.

Sparse gases are well described by Boltzmann equation, but are applicable only forshort-range forces. Speaking about radiation and neitron transport (see Chapters 9, 10,and 11) we obtain linear Boltzmann equations and the problem of cross-sections forthem is still actual. This aspect of the problem we discussed in this chapter and maybe an explanation can be obtained in the field of quantum Hamiltonians.

Correspondence principle quantum Hamiltonians—Kinetic equations can perhapsbe applied to some questions of chemical kinetics. It was explained in detail in thischapter, too.

Navier-Stokes equations for liquids also can be derived from Boltzmann equa-tion, but there are a lot of situations where these basic equations are not applicable(see numerous paradoxes in the 6-th volume of Landau-Lifshits textbook [171]). Forexample, the dependence of viscosity from spatial gradients of velocity was derivedin [200] for big spatial gradients, and it is not Navier-Stokes one. The modern hydro-dinamics theory definitely show that basic equations are not applicable (see commentsin the 6-th volume of the Landau and Lifshitz monograph). Some viscosity phe-nomena, especially for high velocity gradients were solved; see [200], for example.By contrast, discrete Boltzmann equations, especially lattice gases automata (LGA),are widely used today to describe complex properties of liquids. Moreover, LGAscan be efficiently used for other mediums, like hard bodies or magnetic hydrodi-namics. In general, any object where collisions are described in terms of probality,can be represented as LGAs. Evidently such applications should be revised first,to avoid the appearence of extra invariants, see Sections 5.7, 6.6 and Chapter 9 ingeneral.

There are some other examples of applications [9], [10], [44], [95], [250], [285]of the conservation laws in kinetic equations. In [95] we study relations of kineticequations with hydrodinamic approximations. Papers [9], [285] propose an approachto restore formula for entropy from equilibrium states. In particular, paper [9] statesthat this restoration is uniquily defined if the equation has more than one conservationlaw. Results [285], [44] devoted to classification of conservation laws for colisionlesskinetic equations. For the free particle motion in [285] and for an arbitrary externalforces in [44]. Publications [10], [250] considered time-dependent conservation lawsand decreasing functionals. It was shown [250] that Boltzmann equation has eightadditional conservation laws (except five classical); for the Broadwell model [10] theirnumber is continual. Paper [199] continue the study of diagonalization of quantumHamiltonian (they call it Orlov—Vedenyapin diagonalization problem) for the modelof frequency transformation.

13.7 Conclusions

1. This chapter established the relation between quantum Hamiltonians and kinetic equationswhen conservation laws liner in terms of particle numbers are translated into conservationlaws linear in terms of distribution functions. This analogy is valid for chemical kinetics,

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discrete Boltzmann system of equations, triple (or higher order) collisions and coagulation—fragmentation equations, satisfying H- theorem.

2. Why do the systems of equations in mechanics are Hamiltonian (Lagrangian) ones, i.e.,comply with some least action principles? The answer on this question is rather simple, butnontrivial: because elementary particles are being created and annihilated. Speaking aboutcreation and annihilation of particles we can writedown a quantum Hamiltonian, followed byrelated classical one. Moving from quantum to classical representation we have the uniquecorrespondence, while a reciprocal relation is not unique due to uncertanity of quantization.

3. “In the giant manufacture of Nature the principle of entropy plays a role of a director, whichprescribes the form and providing of all bargains. The Energy conservational law plays a roleof only accountant, which calculates debit and credit” (Robert Emden). Which elementaryactions lead to the increase of entropy in nature? Studying the relationship between quan-tum Hamiltonians and Kinetic equations, one can find that the most fundamental act is thecreation and annihilation of particles: the director-entropy in such a manner gambles pair ofdice (according to A. Einstein).

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14 Modeling of the Limit Problemfor the MagneticallyNoninsulated Diode

14.1 Introduction

This chapter is aimed at studying the stationary self-consistent problem of magneticinsulation under space-charge limitation via the asymptotics of the Vlasov-Maxwellsystem. This approach has been introduced by Langmuir and Compton [173] andrecently developed by Degond and Raviart [87], N. Ben Abdallah, P. Degond, andF. M’ehats [35] to analyze the space charge limited operation of a vacuum diode.In a dimensionless form of the Vlasov-Poisson system, the ratio of the typical par-ticle velocity at the cathode to that reached at the anode appears as a small param-eter [87]. The associated perturbation analysis provides a mathematical frameworkto the results of Langmuir and Compton [173], stating that the current flowing throughthe diode cannot exceed a certain value called the Child-Langmuir current. We studythe extension of this approach, based on the Child-Langmuir asymptotics to magne-tized flows [35]. In particular, the value of the space charge limited current is deter-mined when the magnetic field is small (noninsulated diode). Since the arising modelcould not be solved analytically, it is very important to discover its properties in non-insulated and nearly-insulated cases first.

For better understanding of the discussed mathematical problem and especially thecorrespondence of the numerical modeling results with a rising physical effects invacuum diode first we need to introduce the description of how it really works.

The other related important thing is a brief discussion of the physical processesgiving rize to the diode current fluctuations. The better understanding of these proper-ties will be needed for examining current instabilities in the nearly-magnetic insulateddiode. These issues will be discussed at the end of Section 14.6.

The excellent description of this process found in [154] is discussed now.

14.2 Description of Vacuum Diode

The vacuum diode consists of a hot cathode surrounded by a metal anode inside anevacuated enclosure. At sufficiently high temperatures electrons are emitted from the


http://dx.doi.org/10.1016/B978-0-12-387779-6.00014-4

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cathode and are attracted to the positive anode. Electrons moving from the cathode tothe anode constitute a current; they do so when the anode is positive with respect to thecathode. When the anode is negative with respect to the cathode, electrons are repelledby the anode and the reverse current is almost zero (due to the tail of the Maxwelliandistribution of the electrons it is greater than zero). The space between the anodeand the cathode is evacuated, so that electrons may move between the electrodesunimpeded by collisions with gas molecules. If Vf = 0 (no filament voltage) then noemission takes place, the diode may be regarded as a parallel-plate capacitor whosepotential difference is Vp. In this case, the potential distribution in the cathode-platespace is represented by a straight line which joins the points corresponding to cathodepotential Vk = 0 and the plate potential Vp. When the filament voltage rises, the elec-trons leaving the cathode gang up in the interelectrode space as a cloud called a spacecharge. This charge alters the potential distribution. Since the electrons making upthe space charge are negative, the potential in the cathode-plate space goes up, thoughall points remain at positive potential. The vector of the electric field is directed fromthe plate to the cathode, so all the electrons escaping from the cathode make for theplate. In this case, the plate current equals the emission current. One could say theall electrons are being sucked away from the cathode by the anode. This region isknown as the emission-limited region. As the filament voltage is increased, emissionincreases, and so does the space charge. Electrons having low initial velocities aredriven back to the cathode by the negative space charge due to the electrons. Thedensity of the electron cloud near the cathode increases to the point where it forms anegative potential region whose minimum, Vmin, is usually within a few hundredth ortenths of a millimeter of the cathode surface. Thus, there is a high retarding electricfield near the cathode (0< x< xmin); the vector is directed away from the cathodeto the plate. To overcome this field, an initial velocity v0 of the electrons leaving the

cathode should exceed a certain value determined by Vmin, v0 >

√2

e

mVmin.

If the electron is below this value, the electron will not be able to overcome thepotential barrier. It will slow down to a stop, and the field will push it back to thecathode. Accordingly, the retarding field region (from 0 to xmin) contains not onlyelectrons traveling away from the cathode, but also those falling back toward the cath-ode. At a constant filament voltage, a dynamic equilibrium sets in, so that the numberof electrons reaching the plate and the number falling back to the cathode is equalto the number of electrons emitted by the cathode. Therefore, plate current is smallerthan emission current, or the cathode produces more electrons than the anode can.

We know that the minimum signal observable in an electronic circuit is set by thelevel of electrical noise in the system. This is caused by a random fluctuation of voltageand current or electromagnetic fields. Shot noise in a diode is the random fluctuationin a diode current, I, due to the discrete nature of electronic charge. Noise causes thesignal to fluctuate around a given value. The average of these fluctuations are zero,due to their randomness. But the root mean square of the fluctuation is measurable.

Perhaps a discussion of the various types of electrical noise giving rise to degra-dation of the observed signal, would be in order. Electrical noise may persist even

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Modeling of the Limit Problem for the Magnetically Noninsulated Diode 243

after the input signal has been removed from the electronic circuit. This impliesthe existence of a basic limit below which signals are no longer distinguishable. Thesignal-to-noise ratio quantities the observability of an output signal. Hence a measurewhether satisfactory amplification can be obtained is given by this ratio. Thereforein order to ensure the maximum observability of an amplified weak signal we mustensure that the noise power introduced by the circuit devices and components shouldbe as small as possible. External sources of noise can produce electrical interferencein circuits. This may be done by electromagnetic radiation. Examples of this would benarrow-frequency band sources such as radio transmitters, local oscillators and power-supply cables and also broad-band sources such as lightning and fluorescent lamps.Another means by which electrical noise may be induced in an electronic circuit froman external source is electromagnetic induction. Since magnetic fields arise from alter-nating currents, thus by electromagnetic induction corresponding noise signals may beinduced into other circuits or different parts of the same electronic system. In order toreduce such effects we take care in the positioning of critical circuit components totake advantage of the short range of such magnetic fields. Such effects can be greatlyreduced by electrostatic screening (i.e., placing the entire circuit, or at least the sensi-tive portions of it, inside a closed metal box and connecting the box to earth potential).It is important that the total electrostatic screening for a system is earthed at one pointonly this ensures that no large-area circuit earth-loops can exist in which signal mayagain be induced by electromagnetic induction. The main types of internal sources ofnoise present in electronic devices are thermal noise and shot noise. Thermal noise isdue to the random motion of the current carriers in a metal or semiconductor whichincreases with temperature. Thermal noise arises from the random motion of electronsin materials due to their thermal energy of 3kT/2 and therefore occurs even in theabsence of an applied electric field.

Shot noise is due to the random flow of electrons in an electric current and is dueto the particle nature of electric charge. The current flows in a vacuum diode is due toemission of electrons from the cathode which then travel to the anode. Each electroncarries a discrete amount of charge and produces a small current pulse. The averageanode current, Ia, is the summation of all the current pulses. The emission of electronsis a random process depending on the surface condition of the cathode, shape of elec-trodes, and potential between the electrodes. This gives rise to random fluctuationsin the number of electrons emitted and so the diode current contains a time-varyingcomponent. Since each electron arriving at the anode is like a “shot,” the fluctuatingcurrent gives rise to a mean-square shot noise current i2s .

14.3 Description of the Mathematical Model

We consider a plane diode consisting of two perfectly conducting electrodes, a cath-ode (X = 0) and an anode (X = L) supposed to be infinite planes, parallel to (Y;Z).The electrons, with charge −e and mass m, are emitted at the cathode and submit-ted to an applied electromagnetic field Eext = EextX; Bext = BextZ such that Eext ≤ 0

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and Bext ≥ 0. Such an electromagnetic field does not act on the PZ component ofthe particle momentum. Hence, we shall consider a situation where this componentvanishes, leading to a confinement of electrons to the plane Z = 0. The relationshipbetween momentum and velocity is then given by the relativistic relationsV(P)=

Pγm

, γ =

√1+|P|2

m2c2

V= (VX,VY), P= (PX,PY), |P|2 = P2X +P2

Y ,

(14.3.1)

which also can be written

V(P)=5PE(P), (14.3.2)

where E is the relativistic kinetic energy

E(P)= mc2(γ − 1), (14.3.3)

and c is the speed of light. We shall, moreover, assume that the electron distributionfunction F does not depend on Y and that the flow is stationary and collisionless. Theinjection profile G(PX,PY) at the cathode is assumed to be given whereas no electronis injected at the anode. The system is then described by the so-called 1.5 dimensionalVlasov-Maxwell model

VX∂F

∂X+ e

(d8

dX−VY

dA

dX

)∂F

∂PX+ eVX

dA

dX

∂F

∂PY= 0 (14.3.4)

d28

dX2=

e

ε0N(X), X ∈ (0,L), (14.3.5)

d2A

dX2=−µ0JY(X), X ∈ (0,L), (14.3.6)

subject to the following boundary conditions:

F(0,PX,PY)= G(PX,PY), PX > 0, (14.3.7)

F(L,PX,PY)= 0, PX < 0, (14.3.8)

8(0)= 0, 8(L)=8L =−LEext, (14.3.9)

A(0)= 0, A(L)= AL = LBext. (14.3.10)

In this system, the macroscopic quantities, namely, the particle density N, X, and Y arethe components of the current density JX , JY . In the above equations, ε0 and µ0 arerespectively the vacuum permittivity and permeability.

The boundary conditions are justified by the fact that the electric field E =−d8/dXand the magnetic field B=−dA/dX are exactly equal to the external fields when self-consistent effects are ignored (N = JY = 0).

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The 1.5 dimensional model (14.3.4)–(14.3.10) ignores the self-consistent magneticfield due to JX , which would introduce two-dimensional effects, and is only an approx-imation of the complete stationary Vlasov-Maxwell system. In this chapter we espe-cially interested in the case, when the applied magnetic field is not strong enough toinsulate the diode, JX does not vanish and our model can be viewed as an approxima-tion of the Maxwell equations.

In order to get a better insight in the behavior of the diode, we write the model indimensionless variables in the spirit of [87, 88]. We first introduce the following unitsrespectively for position, velocity, momentum, electrostatic potential, vector potential,particle density, current, and distribution function:

X = L, V = c, P= mc, E = mc2,

8=mc2

e, A=

mc

e, N =

ε08

xX2, J =−ecN, F =

N

P2,

and the corresponding dimensionless variables

x=X

X, p=

P

P= (px,py),

v= (vx,vy)=V

V=

p√1+p2

, ε =EE=

√1+p2− 1,

ϕ =8

8, a=

A

A, n=

N

N, j=

J

J, f =

F

F.

The next step is to express that particle emission at the cathode occurs in the Child-Langmuir regime: in such a situation, the thermal velocity VG is much smaller thanthe typical drift velocity supposed to be of the order of the speed of light c. Lettingε = VG/c, we shall assume that

f (0,px,py)= gε(px,py)=1

ε3g(px

ε,

py

ε

), px > 0

where g is a given profile. The scaling factor ε3 ensures that the incoming currentremains finite independently of ε, whereas the dependence on p

εexpresses the fact that

electrons are emitted at the cathode with a very small velocity. We refer to [87, 88] fora detailed discussion of the scaling. The dimensionless system reads

vx∂f ε

∂x+

(dϕε

dx− vy

daε

dx

)∂f ε

∂px+ vx

daε

dx

∂f ε

∂py= 0, (14.3.11)

(x,px,py) ∈ (0,1)×R2,

d2ϕε

dx2= nε(x), x ∈ (0,1) (14.3.12)

d2aε

dx2= jεy(x), x ∈ (0,1). (14.3.13)

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Here nε(x) is a particle density, jεy(x) is a current density in Y direction. The initial andboundary conditions are also transformed

f ε(0,px,py)= gε(px,py)=1

ε3g(px

ε,

py

ε

), px > 0, (14.3.14)

f ε(1,px,py)= 0, px < 0, (14.3.15)

ϕε(0)= 0, ϕε(1)= ϕL, (14.3.16)

aε(0)= 0, aε(1)= ϕL. (14.3.17)

Omitting the complete derivation of the limit system, when ε→ 0, we need tointroduce some notions and notations used ahead.

Definition 14.1. We define θ(x)= (1+ϕ(x))2− 1− a2(x) as an effective potential.

It is readily seen that electrons do not enter the diode unless the effective potential θis nonnegative in the vicinity of the cathode. Therefore, we always have θ ′(0)≥ 0. Thelimiting case θ ′(0)= 0 is the space charge limited or the Child-Langmuir regime. Inview (14.3.16), (14.3.17) (it still hold in the limit ε→ 0), this condition is equivalentto the standard Child-Langmuir condition dϕ

dx (0)= 0.Let θL be the value of θ at the anode θL = (1+ϕL)

2− 1− a2

L. If θL < 0, electronscannot reach the anode x= 1, they are reflected by the magnetic forces back to thecathode and the diode is said to be magnetically insulated. This enables us to definethe Hull cut-off magnetic field, which is the relativistic version of the critical fieldintroduced in [144] in the nonrelativistic case:

aHL =

√ϕ2

L+ 2ϕL.

The diode is magnetically insulated if aL > aHL , and is not insulated if aL < aH

L Indimensional variables, the Hull cut-off magnetic field is given by

BH=

1

Lc

√82

L+2mc2

e8L.

Thus our primary goal is a stugy of noninsulated, or nearly insulated diodes, whichmeans Bext < BH . The complete derivation of the model is given in [35], while weneed only its formal expressions

d2ϕ

dx2(x)= jx

1+ϕ(x)√(1+ϕ(x))2− 1− a2(x)

, (14.3.18)

d2a

dx2(x)= jx

a(x)√(1+ϕ(x))2− 1− a2(x)

, (14.3.19)

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with a corresponding Cauchy and boundary conditions

ϕ(0)= 0, ϕ(1)= ϕL (14.3.20)

dϕ

dx(0)= 0 (14.3.21)

a(0)= 0, a(1)= aL (14.3.22)

Let us recall that the unknowns are the electrostatic potential ϕ, the magnetic potentiala and the current jx (which does not depend on x).

It is to be noticed that the whole construction of this model depends heavily on theassumption that the effective potential is positive. Actually, θ could vanish at somepoints in the diode, leading to closed trajectories and trapped particles.

Apart of heuristic discussions there also could be made some analytical remarksabout the parametric dependences jx, β. In particular, they are

(1+ϕ(x))a′(x)−ϕ′(x)a(x)= β (14.3.23)

2jx√θ(x)− (ϕ′(x))2+ (a′(x))2 = β2 (14.3.24)

The analysis of this equations were made in [35] but the proposed approach do notprovide any information to be immediately used in numerical computations. Never-theless, this relations could be treated as an auxiliary method for verification of any jx,β made.

Vector ( jx, β) hereinafter is usually refered as a parameter vector, depending onthe boundary condition of the problem (14.3.18), (14.3.19). Since the analysis of thecouple arbitrary chosen boundary conditions ϕL, aL is not very useful, we refer to√θ(x) and

√θL especially as a distance measure. The quantities (ϕL, aL) or (ϕL,

√θL)

are algebraic equivalent on R+ to define the boundary conditions, thus we evaluate asets of equally-distant point and refer to them as (z,

√θL), z= ϕL.

Keeping in mind the above remarks we devote Section 14.4 to the analysis of thesolution trajectories, their relation with the lower and upper estimations obtained byA.V. Sinitsyn and better solution approximations. In Section 14.6, we introduce theresults of numerical experiments, describing the properties of the parameter vector fordifferent “distances” θL. The numerical experiments shown that the character of theparameter curves highly depends on the quantity θL.

14.4 Solution Trajectory, Upper and Lower Solutions

Finally, the limit model of magnetically noninsulation diode is described by the sys-tem of two second order ordinary differential equations (14.3.18), (14.3.19) with con-ditions (14.3.20)–(14.3.22).

Let us introduce the definition of cone in a Banach space X.

Definition 14.2. Let X be a Banach space. A nonempty convex closed set P⊂ X iscalled a cone, if it satisfies the conditions:

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(i) x ∈ P, λ≥ 0 implies λx ∈ P;(ii) x ∈ P, −x ∈ P implies x=O, where O denotes zero element of X.

Here ≤ is the order in X induced by P, i.e., x≤ y if and only if y− x is an elementof P.

We will also assume that the cone P is normal in X, i.e., order intervals are normbounded.

In X ≡ {(u,v) : u,v ∈ C1(�),u= v= 0} we introduce the norm |U|X = |u|C1 +

|v|C1 , and the norm |U|X = |u|∞+ |v|∞ in C, where U = (u,v). Here a cone P is givenby P= {(u,v) ∈ X : u≥ 0,v≥ 0 for all x ∈�}. So, if u 6= 0, v 6= 0 belong to P, then−u,−v does not belong. We will work with classical spaces on the intervals I = [a,b],I =]a,b], I = (a,b):C(I) with norm ‖ u ‖∞= max{|u(x)| : x ∈ I};C1(I)=‖ u ‖∞ + ‖ u′ ‖∞;Cloc(I), which contains all functions that are locally absolutely continuous in I. Weintroduce a space Cloc(I) because the limit problem is singular for ϕ = 0. The order≤ in cone P is understood in the weak sense, i.e., y is increasing if a≤ b impliesy(a)≤ y(b) and y is decreasing if a≤ b implies y(a)≥ y(b).

Theorem 14.1 ((comparison principle in cone)). Let y ∈ C(I)⋂

Cloc(I). The functionf is defined on I×R. Let f (x,y) is increasing in y function, then

v′′− f (x,v)≥ w′′− f (x,w) in mean on I, (14.4.1)

v(a)≤ w(a), v(b)≤ w(b) implies v≤ w on I.

For the convenience of defining an ordering relation in cone P, we make a trans-formation for the problem (14.3.18)–(14.3.22). Let F(ϕ,a) and G(ϕ,a) be definedby (14.3.18)–(14.3.22). Then through the transformation ϕ =−u the limit problem isreduced to the form

−d2u

dx2=jx

1− u√(1− u)2− 1− a2

4= F( jx,u,a), u(0)= 0, u(1)= ϕL,

d2a

dx2=jx

a√(1− u)2− 1− a2

4= G( jx,u,a), a(0)= 0, a(1)= aL.

(14.4.2)

We note that all solutions of the initial problem, as well the problem (14.4.2), aresymmetric with respect to the transformation of sign for the magnetic potentiala : (ϕ,a)= (ϕ,−a) or the same (u,a)= (u,−a). Thus we must search only positivesolutions ϕ > 0, a> 0 in cone P or only negative ones: ϕ < 0, a< 0. Thanks to thesymmetry of problem it is equivalently and does not yields the extension of the typesof sign-defined solutions of the problem (14.3.18)–(14.3.22) (respect. (14.4.2)). Oncemore, we note that introduction of negative electrostatic potential in problem (14.4.2)is connected with more convenient relation between order in cone and positiveness ofGreen function for operator −u′′ that we use below.

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Definition 14.3. A pair [(ϕ0,a0), (ϕ0,a0)] is called

(a) sub-super solution of the problem (14.3.18)–(14.3.22) is relative to P, if the followingconditions are satisfied (ϕ0,a0) ∈Cloc(I)

⋂C(I)×Cloc(I)

⋂C(I),

(ϕ0,a0) ∈Cloc(I)⋂

C(I)×Cloc(I)⋂

C(I)(14.4.3)

ϕ′′0 − jx1+ϕ0√

(1+ϕ0)2− 1− a2

4= F(ϕ0,a)≤ 0 in I,

(ϕ0)′′− jx1+ϕ0√

(1+ϕ0)2− 1− a2

4= F(ϕ0,a)≥ 0 in I ∀a ∈ [a0,a

0];

a′′0 − jxa0√

(1+ϕ)2− 1− a20

4= G(ϕ,a0) ≤ 0 in I,

(a0)′′− jxa0√

(1+ϕ)2− 1− (a0)2

4= G(ϕ,a0)≥ 0 in I ∀ϕ ∈ [ϕ0,ϕ

0];

ϕ0 ≤ ϕ0, a0 ≤ a0 in I

and on the boundary

ϕ0(0)≤ 0≤ ϕ0(0), ϕ0(1)≤ ϕL ≤ ϕ0(1),

a0(0)≤ 0≤ a0(0), a0(1)≤ aL ≤ a0(1);

(b) sub-sub solution of the problem (14.3.18)–(14.3.22) is relative to P, if a condition (3.4) issatisfied and

ϕ′′0 −F( jx,ϕ0,a0)≤ 0 in I,

a′′0 −G( jx,ϕ0,a0)≤ 0 in I(14.4.4)

and on the boundary

ϕ0(0)≤ 0, ϕ0(1)≤ ϕL, a0(0)≤ 0, a0(1)≤ aL. (14.4.5)

Remark 14.1. In Definition 14.3 the expressions with square roots we take by modu-lus of effective potential θ(·).

By analogy with (14.4.4), (14.4.5), we may introduce the definition of super-supersolution in cone.

Definition 14.4. The functions 8(x,xai , jx), 81(x,xϕj , jx) we shall call a semitrivialsolutions of the problem (14.3.18)–(14.3.22), if 8(x,xai , jx) is a solution of the scalarboundary value problem

ϕ′′ = F( jx,ϕ,xai)= jx1+ϕ√

(1+ϕ)2− 1− (xai)2,

ϕ(0)= 0, ϕ(1)= ϕL,

(14.4.6)

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and 81(x,xϕj , jx) is a solution of the scalar boundary value problem

a′′ = G(jx,xϕj ,a)= jxa√

(1+ xϕj)2− 1− a2

,

a(0) = 0, a(1)= aL.

(14.4.7)

Here xai , i= 1,2,3 and xϕj , j= 1,2 are respectively, the indicators of semitrivial solu-tions 8(x,xai , jx), 81(x,xϕj , jx) defined by the following way:

xa1 = 0, if a(x)= 0;xa2 = a0, if a= a0 be upper solution of the problem (14.4.7);xa3 = a0, if a= a0 be lower solution of the problem (14.4.7);xϕ1 = ϕ

0, if ϕ = ϕ0 be upper solution of the problem (14.4.6);xϕ2 = ϕ0, if ϕ = ϕ0 be lower solution of the problem (14.4.6).

From Definition 14.4, we obtain the following types of scalar boundary value prob-lems for semitrivial (in sense of Definition 14.4) solutions are

ϕ′′ = F(ϕ,0)= jx1+ϕ√

(1+ϕ)2− 1, ϕ(0)= 0, ϕ(1)= ϕL. (A1)

ϕ′′ = F(ϕ,a0)= jx1+ϕ√

(1+ϕ)2− 1− (a0)2, ϕ(0)= 0, ϕ(1)= ϕL. (A2)

ϕ′′ = F(ϕ,a0)= jx1+ϕ√

(1+ϕ)2− 1− (a0)2, ϕ(0)= 0, ϕ(1)= ϕL. (A3)

a′′ = G(ϕ0,a)= jxa√

(1+ϕ0)2− 1− a2, a(0)= 0, a(1)= aL. (A4)

a′′ = G(ϕ0,a)= jxa√

(1+ϕ0)2− 1− a2, a(0)= 0, a(1)= aL. (A5)

We shall find the solutions of problems (A1)–(A3) for ϕ0 < ϕ0, where ϕ0(xa1),

ϕ0(xa2) are respectively, lower and upper solutions of problem (A1). The solution(ϕ,a) of the initial problem should belong to the interval

ϕ ∈8(ϕ,0)⋂8(ϕ,a0)

⋂8(ϕ,a0),

a ∈81(ϕ0,a)

⋂81(ϕ0,a).

Moreover, the ordering of lower and upper solutions of problems (A1)–(A3) issatisfied

ϕ0(xa1) < ϕ0(xa2) < ϕ0(xa3) < ϕ0(xa2) < ϕ

0(xa1).

We shall seek the solution of problems (A4)–(A5) for a0 < a0. In this case the follow-ing ordering of lower and upper solutions of problems (A4)–(A5)

a0(xϕ1) < a0(xϕ2) < a0(xϕ2) < a0(xϕ1).

is satisfied.

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We go over to the direct study of the problem (14.4.6) which includes the cases(A1)–(A3). Let us consider the boundary value problem (14.4.6) with

F(x,ϕ) : (0,1]× (0,∞)→ (0,∞). (B1)

In condition (B1) for F(x,ϕ) we dropped index ai, considering a general case of non-linear dependence F of x.

We shall assume that F is a Caratheodory function, i.e.,

F(·,s) measurable for all s ∈ R, (B2)

F(x, ·) is continuous a.e. for x ∈]0,1], (B3)

and the following conditions hold

1∫0

s(1− s)Fds<∞. (B4)

∂F/∂ϕ > 0, i.e., F is increasing in ϕ. (B5)

There are γ (x) ∈ L1(]0,1]) and α ∈ R, 0< α < 1 such that

|F(x,s)| ≤ γ (x)(1+ |s|−α), ∀(x,s) ∈]0,1]×R. (B6)

We are intersted in a positive classical solution of equation (14.4.6), i.e., ϕ > 0 inP for x ∈]0,1] and ϕ ∈ C([0,1])

⋂C2(]0,1]). The problem (14.4.6) is singular, there-

fore, condition (B1) is not fulfilled on the interval ϕ ∈ (0,∞) and in this connection,the well-known theorems on existence of lower and upper solution in cone P does notwork. It follows from Theorem 14.1, since F in (14.4.6) is increasing in ϕ, then ϕ < wfor x ∈]0,1], where ϕ and w satisfy the differential inequality (14.4.1).

Theorem 14.2. Assume conditions (B2)–(B6). Then there exists a positive solutionϕ ∈ C([0,1])

⋂C2(]0,1]) of the boundary value problem (14.4.6).

Proof. Let ϕ > 0 is a solution of problem (14.4.6). According to the Theorem 14.1ϕ < w for x ∈]0,1]. Take ε > 0 and consider equation

ϕ′′ε = jx1+ϕε + ε√

(1+ϕε + ε)2− 1− (xai)2

4= Fε( jx,ϕε + ε,xai),

ϕε(0)= 0, ϕε(1)= ϕL.

(14.4.8)

Let w and ϕ are upper and lower solutions of equation (14.4.8) (below, in Proposi-tion 14.1 is shown that such solutions really exist). Hence the theorem on monotoneiterations (see [130]) gives an existence of classical solution ϕε of equation (14.4.8),which satisfies w> ϕε > ϕ for x ∈ (0,1] and is bounded in C. Thus Fε( jx,ϕε + ε,xai)

is bounded and there exists uniform limit limε→0ϕε = ϕ. It follows from the last, if0< η < 1

2 , then limε→0 Fε( jx,ϕε + ε,xai)= F( jx,ϕ,xai) uniformly on [η,1− η] andϕ > 0 for x ∈ [η,1− η].

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Since ϕε is uniformly converged on [0,1], then it implies existence limε→0ϕ′ε(η).

Therefore there exists limε→0ϕ′′ε (x) on the compact subspaces (0,1) and {ϕ′ε} is uni-

formly converged on (0,1) to a differentiable function ϕ′ on [η,1− η]. From the lastit follows that ϕ is twice differentiable on [η,1− η], ϕ′′ = F( jx,ϕ,xai), x ∈ [η,1− η]and u ∈ C([0,1])

⋂C2((0,1]) is a positive solution of the problem (14.4.6).

Remark 14.2. A delicate moment in the proof of Theorem 14.2 is connected with thefinding of a lower ϕ and an upper w solutions for perturbed problem (14.4.8). As alesser solution, we can take the solution of equation (A1) (semitrivial solution ϕ), thenan upper solution will be, for example, maximal solution of equation (A1).

Application of monotone iteration techniques to the equation (14.4.6) gives an exis-tence of maximal solution ϕ(x, jx) such that

ϕ(x,xj)≤ ϕ(x,xj) < w(x) for x ∈]0,1].

Proposition 14.1. Let 0< c≤ jx ≤ jmaxx . Then equation (A1)

ϕ′′ = F( jx,ϕ,0)= jx1+ϕ

√ϕ(2+ϕ)

,

ϕ(0)= 0, ϕ(1)= ϕL

has a lower positive solution

u0 = δ2x4/3, (14.4.9)

if

4δ3≥ 9jmax

x (1+ δ2)/√

2+ δ2 (14.4.10)

and an upper positive solution

u0= α+βx (α,β > 0) (14.4.11)

with

ϕL ≥ δ2, (14.4.12)

where δ is defined from (14.4.10).

Remark 14.3. Square root is taking as√|ϕ(2+ϕ)| in the case of negative solu-

tions. Here u0=−εx is an upper solution, and u0 =−2+ ε is a lower solution

(0< ε < 1). Hence equation (A1) has the negative solution only for 0< ϕL <−2because F(x,−2)=−∞.

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It follows from (14.4.10), (14.4.12) that a value of current is limited by the valueof electrostatic potential on the anode ϕL

jx ≤ jmaxx ≤ F(ϕL). (14.4.13)

Analysis of lower and upper solutions (14.4.9), (14.4.11) exhibits that for δ2= ϕL > 2

and α = β ≤ 1 interval in x between lower and upper solutions is decreased, and forthe large values of the potential ϕL diode makes on regime ϕLx4/3.

Proposition 14.2. Let 0< c≤ jx ≤ jmaxx . Then equation (A4)

a′′ = G( jx,ϕ0,a)= jx

a√(1+ϕ0)2− 1− a2

, a(0)= 0, a(1)= aL

with a lower solution a0 = 0 and an upper solution a0= u0 > 0, conditions (3.14),

(3.16) has an unique solution a(x, jx,c), which is positive, moreover

0≤ aL ≤

√ϕ0(2+ϕ).

Proof. The positive solution of problem (A4) is concave and be found as a solution ofinitial problem with a(0)= 0, a′(0)= c, where c is a shooting parameter. The solutiona= a(x, jx,c) is unique and strongly decreasing in c because the right part of differ-ential equation is decreasing in a. The least nonnegative solution is f (x, jx,0)= 0 and

for 0≤ aL ≤

√ϕ0

L(2+ϕ0L) there exists only one solution and no positive solutions for

other values aL.

Remark 14.4. The problem (A5) is considered by analogy with problem (A4), changeof an upper solution a0

= u0 to a lower a0= u0 one and 0≤ aL ≤

√ϕ0L(2+ϕ0L).

Following to the definition 14.3 and Propositions 14.1, 14.2, solutions of the prob-lems (14.4.6), (14.4.7) we can write in the form

lower-lower (ϕ0,a0)):

ϕ0 = u0 = δ2x4/3, a0 = 0, ϕL ≥ δ

2;

0 1 x

a0

a0

ϕ,a

ϕ0

ϕ0

Figure 14.1 Location of lower (ϕ0,a0) and upper (ϕ0,a0) solutions

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upper-lower (ϕ0,a0):

ϕ0= u0= α+βx, a0 = 0, δ2

≤ ϕL ≤ C, C = max{α,β};

lower-upper (ϕ0,a0):

ϕ0 = u0 = δ2x4/3, a0

= u0, ϕL ≥ δ2, aL ≤

√(u0(2+ u0);

upper-upper (ϕ0,a0):

ϕ0= u0= α+βx, a0

= u0, ϕL ≤ C, aL ≤ a0≤ u0.

14.5 Existence of Solutions for System(14.3.18)–(14.3.22)

In the previous section we demonstrated the existence of semitrivial solutions of sys-tem (14.3.18)–(14.3.22). Here we show the existence of solutions for the completesystem (14.3.18)–(14.3.22) using the following McKenna-Walter theorem.

Theorem 14.3 (see McKenna, Walter [194]). Assume conditions (B1)–(B6). Weassume that there exists the ordered pair (u, u) of lower and upper solutions, i.e.,

u, u ∈ Cloc((0,1])2⋂

C([0,1])2, u≤ u on (0,1]

u(0)≤ 0≤ u(0), u(1)≤ uL ≤ u(1); uL4= (ϕL,aL),

∀x ∈ (0,1] : ∀z ∈ R2,

u(x)≤ z≤ u(x), zk = uk(x);−u′′k (x)≥ hk(x,z)) (14.5.1)

and

∀x ∈ (0,1] : ∀z ∈ R2,

u(x)≤ z≤ u(x), zk = uk(x) :

− u′′k (x)≤ hk(x,z) (14.5.2)

for all k ∈ {1,2}. Then there exists a solution u ∈ C2((0,1])2⋂

C([0,1])2 of theproblem

−u′′ = h(·,u(·)) (0,1]

u(0)= 0, u(1)= uL.

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For keeping of ordering of lower and upper solutions in Theorem 14.3 (in cone P)we write differential inequalities (14.5.1), (14.5.2) in the following form

∀z ∈ [v(x),w(x)], z1 = w1(x):

± w′′1(x)T± F1(w1(x),z2)

∀z ∈ [v(x),w(x)], z1 = v1(x):

± v′′1(x)S± F1(v1(x),z2)

∀z ∈ [v(x),w(x)]; z2 = w2(x):

± w′′2(x)T± F2(z1,w2)

∀z ∈ [v(x),w(x)]; z2 = v2(x):

± v′′2(x)S± F2(z1,v2).

Remark 14.5. The change of signs with (+) to (−) in differential inequalities is con-nected with adjustment of signs and ordering (≤) of lower (upper) solutions of system(14.3.18)–(14.3.22) in Definition 14.3 and lower (upper) solutions in Theorem 14.3.

From the last relations we obtainw′′(x)= F1(w1(x),0)≤ F1(w1,z2)

v′′1(x)≥ supz2

F1(v1(x),z2),

w′′2(x)≤ F2(z1,w2)

v′′2(x)≥ supz1

F2(z1,v2).

From inequality v′′2(x)≥ supz1F2(z1,v2), we get estimations to the value of mag-

netic field on the anode aL

aL ≤jx2≤

jmaxx

2≤F(ϕL)

2(14.5.3)

taking account of (14.4.13) and θL > 0. Under realization of estimation (14.5.3) thediode works in noninsulated regime, moreover, the value aL is limited by value ofelectrostatic potential on the anode ϕL with a critical value ϕL = 2. In increasing ofmagnetic potential aL the diode transfers in isolated regime that leads to more compli-cated problem with free boundary.

Thus, we have the following main result of this paper.

Theorem 14.4. Assume conditions (B2), (B3), (B6), and inequalities (14.4.10),(14.4.13) and

aL ≤jx2≤

jmaxx

2≤F(ϕL)

2

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fulfilled. Then the problem (14.3.18)–(14.3.22) possesses a positive solution in cone Psuch that{

ϕ′′0 ≥ jxF(ϕ0,z2), z2 ∈ [0,ϕ0]

(ϕ0)′′ ≤ jxF(ϕ0,z2), z2 ∈ [0,ϕ0],

{a′′0 ≥G( jx,z1,a0), z1 ∈ [ϕ0,ϕ

0]

(a0)′′ ≤G( jx,z1,a0), z1 ∈ [ϕ0,ϕ

0],

where ϕ0 = δ2x4/3 is a lower solution of problem (A1), ϕ0

= α+βx (α,β > 0) is anupper solution of problem (A1) with condition ϕL ≥ δ

2; a0 = 0 is a lower solution ofproblem (A4) with condition 0≤ aL ≤

√ϕ0(2+ϕ0).

Theorem 14.4 may be used to the construction of the minimal and maximal solutionof (14.3.18)–(14.3.22) on the basis of monotone-iteration method in Heikkila [130].

14.6 Analysis of the Known Upper and Lower Solutions

Up to this moment the analytical solution of the ODE system defined by (14.3.18),(14.3.19) with respect to the conditions (14.3.20)–(14.3.22) is unknown. The onlynown result partially describing the form of the solution trajectory was given 14.1, seealso [261]. According to it, both solution trajectories are bounded by the upper andlower solutions

yUP(x)= kx+ b, k,b> 0 (14.6.1)

yLOW(x)= c2x43 (14.6.2)

Using the boundary conditions (14.3.20), (14.3.22) one can obtain a quite good solu-tion trajectory estimations

c2ϕLx43 ≤ ϕ(x)≤ ϕLx, 0≤ a(x)≤ aLx

defined on x ∈ [0, 1]. Here and everywhere we assume boundary conditions ϕl, aL

correctly defined, i.e. θL > 0. Looking forward and leaving the discussion of numer-ical solution methods for next the sections, here we provide some numerical solu-tion trajectory examples both for ϕ(x) and a(x) evaluated for different boundaryconditions.

The straightforward analysis of the trajectories Figures 14.2–14.4 shows that thelower solutions obtained in Section 14.4 could be made significantly better and theupper solutions are exactly ϕLx and aLx. The lower solution obviously could be writ-ten as

yLOW(x)= y(1)xγ, γ > 1. (14.6.3)

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00.20 0.4

x

ϕ (x) a (x)

x0.6 0.8 1 0.20 0.4 0.6 0.8 1

0.2

0.4

0.6

Phi

(x)

0.8

1

0

0.2

0.4

0.6

a(x

)

0.8

1

Figure 14.2 Numerical solution for ϕL = aL = 1; numerical integration error εϕ =

2.4611389315421e− 17, εa = 6.13116328540553e− 17; estimated jx = 0.534075023488271,da

dx(0)= 0.879738089874635. Function ϕ(x): upper solution y= x and lower solution

y=7

10x

43 . Function a(x): upper solution y= x and lower solution y= x

43 .

00.20 0.4

x

ϕ (x) a (x)

x0.6 0.8 1 0.20 0.4 0.6 0.8 1

2

4

6

Phi

(x)

8

0

0.5

1

1.5a(x

)

2

2.5

3

Figure 14.3 Numerical solution for ϕL = 8, aL = 3; numerical integration error εϕ =

3.27429056090622e− 16, εa = 1.19262238973405e− 17; estimated jx = 8.93859989164142,da

dx(0)= 1.72776197665836. Function ϕ(x): upper solution y= 8x and lower solution y= 5x

43 .

Function a(x): upper solution y= 3x and lower solution y=5

2x

43 .

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00.20 0.4

x

ϕ (x) a (x)

0.6 0.8 1 0.2 0.4x

0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

Phi

(x)

0.3 0.8

0.6

0.4

0.2

00

a(x

)

Figure 14.4 Numerical solution for ϕL = 0.3, aL = 0.8; numerical integration errorεϕ = 1.71574993795831e− 17, εa = 5.97937498125756e− 17; estimated jx =

0.0761231763035411,da

dx(0)= 0.759092882499624. Function ϕ(x): upper solution y= 0.3x

and lower solution y= 0.18x43 . Function a(x): upper solution y= 0.8x and lower solution

y= 0.8x43 .

2

20 40z

60 80

4

6

8

10

12

J x, be

ta

14

16

18

Figure 14.5 jx,β parameter curves for θL = 1.

Here y(1) is ether ϕL or aL. The value of the parameter γ depends only on ϕL, aL andcould be found numericaly.

Since it is much easy to view the modeling results, first we draw them in a picturesfor (z,

√θL)= (z,1):

a) Figure 14.5 assume z= ϕL = 2, . . . ,30,35,40,45,50,60,70,80;b) Figure 14.6 assume z= ϕL = 0.45,0.5.1,1.25.1.5,1.75,2,2.5,3.

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0.5

1

1.5

Jx, be

ta

2

2.5

z0.5 1 1.5 2 2.5 3

Figure 14.6 jx,β parameter curves for θL = 1.

The upper curve correspond to parameter jx at Figure 14.5 and to β at Figure 14.6;the lower—to parameter β at Figure 14.5 and to jx at Figure 14.6.

This two figures were provided separately for two purposes:

a) β trajectory seems to keep the same character over the whole interval. This character islikely to be (ax)b, b< 1;

b) The jx trajectory character is different for z≤ 2.5 and z> 2.5. The right branch is theconvex function, but at the left the sign of the j(2)x (ϕL,aL) is changed. The corresponding

approximation curve could be similar to axb

xc+ d, for example.

14.7 Conclusions

One of the main conclusions to be made is the fact that a statement of limit problemdoes not comply with a Child-Langmuir regime that a current density jx is saturatedin the case of noninsulated and nearly-insulated diode. Moreover, with a respect to therough numerical approximations made, the current density jx will infinitely grow ifthe voltage applied to the diode also grows. On the other hand, the experimental datashow that jx grow faster for noninsulated diode than for a nearly-insulated one. Thiscould be seen as a preliminary numerical proof that a Child-Langmuir regime couldbe achieved only in magnetically insulated diode.

The second conclusion refers to the nearly-insulated diode. According to the earliermade comments, the discovered properties described by a limit problem in mathemat-ical statement fully comply with their physical expectations described in introductionsection. Jointly with a first conclusion, it characterize the obtained limit model as areliable one, that comply with the physical processes, underlying the thermo-vacuumdiode with a plane cathode and anode.

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15 Generalized Liouville Equationand Approximate OrthogonalDecomposition Methods

15.1 Introduction

At the beginning we have to outline some aspects, concerning classic Liouvilletheorem (Liouville 1838 [178]) and equation for the ODE system

x= X(x, t), x(t0)= x0∈ Rn (15.1.1)

where x ∈ Rn; J = {t : t0 ≤ t <+∞}; Xi(x, t) ∈ C(1,1)xt (G); G=�× J; �⊆ Rn—abounded domain. If the named conditions are fulfilled, then at arbitrary moment t0a unique solution x(t)= x(x0, t0, t) of the Cauchy problem (15.1.1) passes througheach point x0

∈ Rn [162, 233].It is well known (see Nemitskij, Stepanov 1949 [215], Kaplan 1953 [153]), that the

ODE system (15.1.1) has a corresponding Liouville equation:

∂

∂tf (x, t)= Lf (x, t), f (x, t0)= f0(x). (15.1.2)

Here

L· = −n∑

i=1

∂

∂xi[Xi(x, t)·]=−div[X(x, t)·] (15.1.3)

is a Liouville operator. Assume f (x, t) ∈ L2(R), t ∈ J and suppose L acting like

L : C∞0 (Rn)→ L2(Rn). (15.1.4)

Here function f0(x) is defined below:

f0(x)≥ 0, f0(x) ∈ C∞0 (Rn),

∫Rn

f0(x)dx= 1. (15.1.5)


http://dx.doi.org/10.1016/B978-0-12-387779-6.00015-6

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l χ(x, t)= divX(x, t)—divergence of vector field for ODE (15.1.1);l D(x(x0, t0, t), t)= det

[∂x(x0,t0,t)

∂x0

]—Jacobian of inverse transition x0

→ x(x(x0, t0, t);

l S(x, t)= det[∂x0(x,t,t0)

∂x

]—Jacobian of transition x(x0, t0, t)→ x0;

l χ(x(x0, t0, t), t)—divergence of vector field for ODE (15.1.1) calculated along the solutiontrayectory x(t)= x(x0, t0, t).

As usual, we treat the Gibbs ensemble of represented points (Gibbs 1902 [109])for the system of equations (15.1.1) as a set of the identical systems (15.1.1) havingdifferent initial states.

Let �t0 ⊂� be a compact Lebesgue mes �t0 measure set filled by a Gibbsensemble of represented points for the ODE system (15.1.1) at the moment t = t0.Each of the represented points x0

∈�t0 , moving along with the ODE (15.1.1) trayec-tories, is shifted to a new state x(x0, t0, t)= T(t, t0)x0

∈�t ⊂� starting from momentt0 to t. Here T(t, t0) is a shift operator along the ODE system (15.1.1) trayectories(Krasnoselskij 1966 [162]);�t = {x(x0, t0, t)= T(t, t0)x0 : x0

∈�t0}—an image of theset �t0 according to ODE system (15.1.1). It means that �t = T(t, t0)�t0 . Let mes �t

is a Lebesgue measure of the set �t ⊂�n.

Function f0(x)which comply with conditions (15.1.5) can be considered as a distri-bution density function for the Gibbs ensemble of represented points from�t0 , system(15.1.1). The current value of distribution density function f (x, t) ∈ L2(Rn), t ∈ J, isdefined by the initial conditions of the Cauchy problem (15.1.2), (15.1.3). It describesthe state of the Gibbs ensemble of represented points for the ODE system (15.1.1) inthe image �t of the set �t0 .

To indroduce a uniqueness and existence results we need to introduce the followingassumption:

ASSUMPTION (A) holds for the ODE system (15.1.1), if the solution x(t)=x(x0, t0, t) is nonlocally continuable (Krasnoselskij 1966 [162]) on J for all repre-sented points x0

⊂�t0 and is kept in � ∀t ≥ t0.

Definition 15.1. We call a function f (x, t) ∈ L2(Rn) a classical solution of the Cauchyproblem (15.1.2) with operator (15.1.3) acting according to (15.1.4), if the substitutionof this function f (x, t) into the Liouville equation (15.1.2) turns it into identity.

Then theorem holds [246].

Theorem 15.1. First. Let assumption (A) holds for the ODE system (15.1.1).Second. The corresponding ensemble of Gibbs represented points has an initial

distribution density function f0(x) satisfying conditions (15.1.5) in the compact set�t0 ⊂�.

Third. Let �t = {x(x0, t0, t)= T(t, t0)x0 : x0∈�t0} be an image of the set �t0

defined according to system (15.1.1) and D(x(x0, t0, t), t) 6= 0.Hence the shift operator T(t, t0) along the solution trayectories (15.1.1) defines a

set homeomorphism �t0 ⊂� onto the set �t ⊂�; There exists the unique solution ofthe Cauchy problem (15.1.2)–(15.1.4) ∀t ∈ J and it complies with conditions

f (x, t)≥ 0, f (x, t) ∈ C∞0 (Rn),

∫Rn

f (x, t)dx= 1, (15.1.6)

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Generalized Liouville Equation and Approximate Orthogonal Decomposition Methods 263

f (x(x0, t0, t), t)= f0(x0)e

[−

t∫t0

χ(x(x0,t0,t),t)dt

]= f0(x

0)/D(x(x0, t0, t), t), (15.1.7)

f (x, t)= f0(p(x, t, t0))e

[−

t∫t0

χ(x(p(x,t,t0),t0,τ ),τ )dτ

]= f0(p(x, t, t0))S(x, t). (15.1.8)

If we denote L—Liouville operator (15.1.3); p(x, t, t0)= T−1(t, t0)x= x0, thenthere hold the relations introduced below:

d

dtlnD(x(x0, t0, t), t)= χ(x(x

0, t0, t), t), D(x(x0, t0, t), t)∣∣t=t0= 1, (15.1.9)

∂S(x, t)

∂t= LS(x, t), S(x, t)

∣∣t=t0= 1, (15.1.10)

mes �t =

∫�t0

e

[t∫

t0

χ(x(x0,t0,t),t)dt

]dx0, (15.1.11)

mes �t =

t∫t0

∫�t

χ(x,τ )dxdτ +mes �t0 . (15.1.12)

The above theorem 15.1 has some other interpretations. For example, Nemitskij,Stepanov 1949 [215]; Zubov 1982 [323] treated function ρ(x, t) satisfying Liouvilleequation (15.1.2), (15.1.3) as a kernel or the density of the integral invariant. Resolvingn equations x= x(x0, t0, t) with respect to n initial conditions x0, we have

x0= T−1(t, t0)x≡ p(x, t, t0). (15.1.13)

Here functions p(x, t, t0) are the first n independent integrals of the ODE system(15.1.1). The transformation mentioned above could be made since a transitioninvolved by shift operator T(t, t0) is homeomorphic and conditions on the implicitfunction are hold.

Theorem 15.2 (Zubov 1982, [323]). Assume

1. Let the solution x= x(x0, t0, t) of the system (15.1.1) exists for t ∈ (−∞,+∞), t0 ∈(−∞,+∞), x0

∈ Rn;2. Let vector function (15.1.13) exists for t ∈ (−∞,+∞), t0 ∈ (−∞,+∞), x ∈ Rn,

then each nonnegative function ρ0(x) 6= 0 on x ∈ Rn is continuously differentiable byall arguments and possesses a unique nonnegative solution ρ(x, t) of the equation

∂ρ(x, t)

∂t+5 · [X(x, t)ρ(x, t)]= 0,

such that ρ(x, t)= ρ0(x) at t = t0. Function ρ(x, t) is a kernel of integral invariant forthe system (15.1.1) also.

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Liouville theorem and equation play a great role (Hinchin 1943 [133]; Krylov1950 [164]; Bogolubov [49]; Prigozhin 1964 [234]) in statistical mechanics. Pro-viding a statistical justification of the principles and by discovering the structure ofthe multiple body and/or multiple process systems tending to the equilibrium state.Invariant measure Liouville theorem (Arnold, Kozlov, Neıshtadt 1985 [14]) is thebasis for qualitative studying methods of the n–body problem (Hilmi 1951 [132]).Liouville equation was taken as an initial point for the ergodicity theory (Cornfeld,Fomin, Sinai 1980 [78]); for the kinetic theory of irreversible processes; in derivationof Vlasov-Maxwell (VM) integro-differential equations (Vlasov 1950 [305]). Exactlyspeaking, Vlasov equation [305] could be derived from Liouville equation for thecharged particles distributuion function, neglecting particle correlations and suppos-ing many-particle distribution function as a direct product of proper single-particledistribution functions. An application of Liouville theorem for studying VM equa-tion also could be found in (Maslov, Fedoryuk 1985 [191]; Lewis, Barnes, Melendez1987 [176]; Horst 1990 [143]) and some recent papers. An infinite dimensional for-mal Hamiltonian approach for the infinite dimensional VM system was developed by(Morrison 1980 [211]; Marsden, Weinstein 1982 [186]). The mentioned papers intro-duce Poisson bracket evaluation technique for the VM system and prove that it is aninfinite dimensional Hamiltonian system, i.e. could be written as Liouville equation.

It seems that Bogolubov was the first mathematician who introduced the “classical”representation of the Liouville equation

∂

∂tf (q,p, t)= [H(q,p, t), f (q,p, t)], f (q,p, t0)= f0(q,p) (15.1.14)

to describe the probabilistic properties of the canonical Hamilton equation system

qi =∂

∂piH(q,p, t), pi =−

∂

∂qiH(q,p, t), qi(t0)= q0

i , pi(t0)= p0i

(15.1.15)

with an arbitrary initial states distributed in R2n phase space.Here q,p ∈ Rn—are the generalized coordinate and generalized conjugate impulse

vectors correspondingly; t ∈ R= (−∞,+∞); H(q,p, t) : G→ R, G⊂ R2n+1—Hamilton function from C2 with respect to coordinates q,p;

[H, f ]=n∑

i=1

(∂H

∂qi

∂f

∂pi−∂H

∂pi

∂f

∂qi

)is the Poisson bracket; f0(q,p) and f (q,p, t) are the Gibbs ensemble of representedpoints [109] for the system (15.1.15) in R2n. This functions satisfy the probabilityconditions

f0(q,p)≥ 0,∫

R2n

f0(q,p)dqdp= 1,∫

R2n

f (q,p, t)dqdp= 1.

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The “classical” equation (15.1.14) is a particular case of the so-called generalizedLiouville equation (15.2.1), which deals with compressible or dissipative dynamics.This equation is discussed futher. One of the most reperesentative modern digestsconcerning generalized Liouville equations was made by Ezra [102]. An interestedreader also can find there a wide list of basic and recent bibliographic references.

At present Liouville theorem and generalized Liouville equation are widelyused for proving existence theorems (Povzner [233]), optimal control synthesis forbeam trajectories (Ovsiannikov [97, 222]); stability researches (Fronteau [105];Rudykh [245]; Zhukov [322]); dynamic properties analysis (Misra [207]; Steeb [268];Fronteau [106]; Rudykh [246]); qualitative investigation of dynamic systems(Cornfeld, Fomin, Sinai [78]); discovering the stochastic behavior of dynamic systems(Sinai [260]) and molecular dynamics with applications to chemistry (Tuckerman,Martyna [275]. Look for some other developments of Tuckerman’s group inSection 15.3).

Due to the importance of the problem, our primary goal becomes the constructionof iterative analytical integration methods. In Section 15.2, we provide a completeproblem statement and introduce some remarks about analytical iterative solutions.In Section 15.3 we introduce some of the latest results to be discussed and com-pared later in the text. Section 15.4 contains new results concerning the otherclassical approach for evaluating asymptotical orthonormal decompositions—eigen-value/eigenvector operator decomposition. Section 15.5 is a reminder for the resultspublished in [99] necessory for the numerical modeling presented in the next section.

15.2 Problem Statement

In this chapter, we consider a method of approximate integration for Cauchy problemof the generalized Liouville equation [50, 225]

∂

∂tf (q,p, t)= Lf (q,p, t), f (q,p, t)|t=0 = f0(q,p) (15.2.1)

corresponding to the autonomous system of quasicanonical Hamilton equations

qi =∂

∂piH(q,p), pi =−

∂

∂qiH(q,p)+Q∗i (q,p), (15.2.2)

qi(t)|t=0 = q0i , pi(t)|t=0 = p0

i .

An additive inclusion of the nonpotential term Q∗(q,p) allows to construct propermeasure for existence theorem more easily. Definitely speaking,

L· = [H(q,p), ·]−n∑

i=1

∂

∂pi{Q∗i (q,p)·} (15.2.3)

is a Liouville operator, f (q,p, t) ∈ L2(R2n). Here the∂

∂pi{Q∗i (q,p)·} term corresponds

to the divergent criterium of the solution stability–instability issue.

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Assume the transition rule as given below:

L· =D(L)= C∞0 (R2n)→ R(L)= L2(R2n). (15.2.4)

q,p ∈ Rn are a vector of generalized coordinates and generalized conjugate impulsevector; H(q,p) ∈ C(2,2)qp (R2n)—Hamiltonian of the system; Q∗i (q,p) ∈ C(1,1)qp (R2n)

nonpotential generalized forces; χ(q,p)4=∑n

i=1∂∂pi

Q∗i (q,p) divergence of vectorfield for the system (15.2.2); [·, ·] is Poisson bracket; f0(q,p), f (q,p, t) are the initialand current values of probability density function for the Gibbs ensemble of repre-sented points in the system of equations (15.2.2) in R2n;∫

R2n

f0(q,p)dqdp= 1, t ∈ R+ 4= {t : 0≤ t ≤∞}.

The equation (15.2.1) and corresponding operator (15.2.3), (15.2.4) are widelystudied. Research papers [225, 226] seem to be the most earlier ones. In the semigrouptheory the Cauchy problem (15.2.1) with operator (15.2.3) acting as C∞0 (�)→ L2(�),�⊂ R2n was studied in papers [244]. In particular, there was proved a existence anduniquines theorem for the problem (15.2.1). But this theorem can not provide an effec-tive algorithm to the analytical problem solution. Such development is of a greatpractical importance for the complex multidimensional generalized Liouville equa-tion (15.2.1) and it’s solution f (q,p, t).

To obtain a solution one can integrate (15.2.1) numerically [62], but it is hardlyacceptable, since f (q,p, t) complitely describes the system properties dependendingon q,p, t. Thus it is more valuable to obtain such a relationship in analitical form.Moreover, a numerical integration takes a lot of computational time even for smalldimensions n and advanced integration methods as quasi-Monte Carlo (q-MC) on thelow discrepancy lattices.

On the other hand, since the probability density function f (q,p, t) completelydescribes a system (15.2.2), we need it to describe the time dependences of the meanand dispersion for generalized coordinate and impulse vectors q,p also. Using thisprior statistical knowledge we can try to reduce the dimension of the problem, keepingthe statistical properties unchanged, applying specific group renormalization method.Or to replace the initial problem with an equivalent one (also of smaller dimention).The term “equal” should be defined very carefully for each problem type.

Nevertheless, this idea could be applied more effectively if we want to study thelocal means, for example. This method, i.e., modeling the mean properties of thedynamic systems, combined with advanced Monte-Carlo (MC) and q-MC numericalintegration techniques proved to be very usefull for the group of Berkley scientistslead by Chorin (see articles [71]–[73] for example). Called “Stochastic optimal pre-diction” in general, it is compatible with Hamiltonian formalism and becomes veryusefull for a preliminary research.

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Therefore we are interested in evaluating f (q,p, t) as analytical expression:converging series which coefficients could be simply evaluated and the conver-gence properties could be analytically studied. If the orthonormal function system{hk(q,p)}∞k=0 ∈ L2(R2n) is apriory known or already constructed, the solution of theinitial problem (15.2.1) is translated into the solution of the infinite system of dif-ferential equations to determine the dk coefficient values in expansion f (q,p, t)=∑∞

k=0 dk(t)hk(q,p). On the first glance this approach seems to be pure analytical, butit does not void the applicability of numerical methods, especially numerical integra-tion methods. One of the main joints between them concerns the convergence study.Since the majority of applications can be studied only in terms of truncated seriesf (q,p, t)≈

∑Nk=0 dk(t)hk(q,p), and as it will be shown later—the formal convergence

criteria are hard to check, just taking care of formal condition (15.1.5) becomes a realheadache. But it can be a unique available computational criteria.

On the other hand, we can find the solution f (q,p, t) of the problem (15.2.1) on thebasis of iterative operator method in the small time space over the apriori constructedsystem of orthonormal functions {9k(q,p)}∞k=0 ∈ L2(R2n):

f (q,p,τ )=∞∑

k=0

ak(τ )9k(q,p).

A proposed approach makes it possible to bypass the solution of the infinite systemof differential equations and establish some convergence propositions. The methodintroduced below could be treated as a combination of the mentioned approaches forthe equation (15.2.1).

15.3 The Overview of Preceeding Results

Here we state some earlier results for the problem (15.2.1). Only some of them werepublished before internationally. Since the approaches used there are rather simple, wedo not focus on their details.

15.3.1 An Approach of M.E. Tuckerman Group on the ClassicalMechanics of Non-Hamiltonian Systems

Here we will outline some basic results obtained by Professor M.E. Tuckerman andhis workgroup (see [276, 277] for example) on the generalized Liouville equation andnon-Hamiltonian dynamical systems.

It is known that Hamiltonian flow preserve the measure of phase space treatedas a Euclidean manifold. It differes for non-Hamiltonian flow, since measure is notpreserved in general case. Thus we could state another question: “Is there an invari-ant measure keeping phase volume unchanged for non-Hamiltonian flows?” A partialanswer could be found in the discussed articles. It is shown that such a key concepts ofHamiltonian systems such as “invariant measure” and “continuity” can be generalized

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to the non-Hamiltonian case by a proper treatment of the geometry of the phase spaceand that an invariant measure on the phase space manifold can be derived. Thus, weintroduce a general Riemannian manifold to derive the generalized Liouville equationfor non-Hamiltonian systems of the type

∂

∂t( f√

g)+∇ · ( f√

gx)= 0, (15.3.1)

with

xi= ξ i(x, t) (15.3.2)

a non-Hamiltonian dynamical system for the evolution of the n coordinates x=x1, . . . ,xn with initial values x1

0, . . . ,xn0. The n coordinates describe a point P of an

n-dimensional Riemannian manifold G with metric G. The phase space must be treatedas a general Riemannian manifold with arbitrary curvature, and the volume n-form,which determines the volume element in an arbitrary coordinate system, should beexpressed as w=

√gdx1∧ . . .∧ dxn. The general statement of Liouville’s theorem for

non-Hamiltonian system becomes

f (xt, t)e−w(xt)dx1

t . . .dxnt = f (x0,0)e

−w(x0)dx10 . . .dxn

0 (15.3.3)

with√

g= e−w(x).Right now we are able to establish the correspondence between (15.3.1)–(15.3.3)

and the problem stated is this paper. Liouville equation (15.1.3) for the ODE sys-tem (15.1.2) coincides with (15.3.2) for

√g= 1, and the generalized Liouville equa-

tion (15.2.1) for Non-Hamiltonian system (15.2.2) is a partial case.Hence the Liouville theorem for non-Hamiltonian dynamics allows us to find more

than just an invariant measure construction. We are able to derive the initial distribu-tion function (15.3.3) f0 used for Liouville operator Lf0. All that we need is a Liouvilleoperator Lf0.

While discussing the possibilities one should have note invariant measure construc-tion for the generalized Liouville equation written for a real non-Hamiltonian systemusually appears to be very difficult. Hence we take an initial distribution function firstwithout paying attention on the existance of such invariant measure. Second, we eval-uate the Poisson bracket or Liouville operator to study the dynamics of the initialHamiltonian or Non-Hamiltonian system.

Moreover, Tukermann’s approach is not concerned with the solution of theLiouville equation. One just use the generalized law on the conservation of the phazevolume. It allows to construct the invariant measure without the proper solution of theLiouville equation. Differentely speaking, we are dealing with statistical ensemblesfor the non-Hamiltonian systems, using only the conservation laws.

Such an approach used for derivation of (15.3.2) is based on the differential geom-etry and is well known for ages. The new term is its application for the nonequilibriumdissipative thermodynamics.

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15.3.2 Small–Time Parameter Method

Considering the formal asymptotic solution of Cauchy problem (15.2.1) by the methodof the small time space, proposed in [223] and defined by mapping τ = 1− e−st, s> 0

we transform the initial infinite time interval R+ in a small, finite one J4= {τ : 0≤

τ < 1}. This technique is quite general and was applied to Liouville equation byG. Rudykh and A. Sinitsyn at the earlier 1980s. This results never were translatedinto english earlier, but still are valuable.

The small–time transformation is also well known to the applied mathematitians,becouse it suits for numerical integration over the semi-infinite intervals. The obvi-ous benefit for a such kind of transform lies in power series expansion over time scale.Using finite interval [0, 1] one has |τ k

| ≤ 1, k = 0,1, . . ., thus all attention while study-ing convergence of (15.3.5) is payed to the bounds of series coefficients.

In our particular case a transformed Cauchy problem becomes

(1− τ)∂

∂τf (q,p,τ )=

1

sLf (q,p,τ ), f (q,p,τ )|τ=0 = f0(q,p). (15.3.4)

A solution of the Cauchy problem (15.3.4) in the small time space was studied in theform of asymptotic expansion

f (q,p,τ )=∞∑

k=0

fk(q,p) · τk. (15.3.5)

Substituting (15.3.5) into (15.3.4) and equating the coefficients at the same τ orders,one obtain

fk(q,p)=k− 1

kfk−1(q,p)+

1

sk[H(q,p), fk−1(q,p)]−

−1

sk

n∑i=1

∂

∂pi{Q∗i (q,p) · fk−1(q,p)}.

Hence

fk(q,p)=1

skL

[k−1∏r=1

(1+

1

srL

)]f0(q,p), k = 2,3, . . . (15.3.6)

with

k−1∏r=1

(1+

1

srL

)= ak−1+

1

sak−2L+ ·· ·+

1

sk−1a0Lk−1

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where a0 = 1/(k− 1)!, ak−1 = 1, and

ai =(−1)i

i!a0 ·

∣∣∣∣∣∣∣∣∣∣∣∣

s1 1 0 0 . . . 0s2 s1 2 0 . . . 0s3 s2 s1 3 . . . 0. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

si si−1 si−2 si−3 . . . s1

∣∣∣∣∣∣∣∣∣∣∣∣, i= 1,2, . . . ,k− 1,

si =

k−1∑l=1

zil, zl =−l.

So, the expression (15.3.6) can be written in brief form

fk(q,p)= b1kLf0(q,p)+ b2kL2f0(q,p)+ ·· ·+ bkkLkf0(q,p). (15.3.7)

Here bjk =ak−j

ks j> 0,

∑kj=1 bjksj

= 1 for k = 1,2, . . ..

The recurrent relation (15.3.7) express fk(q,p) in terms of initial probability densityfunction f0(q,p). Consequent evaluation of fk(q,p) terms by equation (15.3.7) withfurther backward substitution into (15.3.5) gives a formal asymptotic solution of theCauchy problem (15.3.4). Using the definition of a small time parameter method, onecan derive the solution f (q,p, t) of the initial Cauchy problem (15.2.1).

To study the convergence properties of the developed asymptotic solution, we needone assumption to be made. Let there exist such constant c> 0, that ∀k = 1,2, . . . thereholds the following inequality

‖Lkf0(q,p)‖L2(R2n) ≤ ck· ‖ f0(q,p)‖L2(R2n). (15.3.8)

Honestly speaking, this condition is rather rigid and will be disscused further.Nevertheless, such a formal assumptions are typical both in pure and applied math.

A general example is a Kantorovitch’s lemma on the local convergence of Newtonmethod. Everyone knows that there exist a set of conditions to guaranty the conver-gence, but it is hard to check them. Another remark concerns the “width” of this localdomain. It could happen that an initial point selected upon this conditions could betreated as a perfect approximation in hardware computatons.

Here and that follows we denote Lkf0(q,p)= L(Lk−1f0(q,p))—an embeddedLiouville operator.

Remark 15.1. Since the generalized Liouville operator is unbounded, it’s embeddeddegree Lk has a restricted definition domain and in general, more restricted then adomain of the initial operator L.

Proposition 15.1. Let for the generalized Liouville operator L defined by formula(15.2.3) acts according to (15.2.4); The inequality (15.3.8) and condition cτ/s< 1 are

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hold. Then the series∑∞

k=0 fk(q,p)τ k is weakly convergent to the element f (q,p,τ ) ∈L2(R2n)—a solution of the Cauchy problem (15.3.4).

Besides, there holds an estimation∥∥∥∥∥∞∑

k=0

fk(q,p)τk

∥∥∥∥∥L2(R2n)

≤s

(1− τ)(s− cτ)· ‖ f0(q,p)‖L2(R2n). (15.3.9)

Proposition 15.2. Let the generalized Liouville operator (15.2.3) act according to(15.2.4); The inequality (15.3.8) and condition c/s< 1 are satisfied. Then the series∑∞

k=0 fk(q,p)τ k strongly converges to the element f (q,p,τ ) ∈ L2(R2n)—the solutionof Cauchy problem (15.3.4) and the relation

( fk, fn)L2(R2n) ≤c2

(s− c)2· ‖ f0‖L2(R2n) = B

holds.

Remark 15.2. If the inequality c/s< 1 holds, then the inequality cτ < 1 holds either.Differently speaking, if series

∑∞

k=0 fk(q,p)τ k converges strongly to the elementf (q,p,τ ) ∈ L2(R2n), then it is weakly convergent also.

Let the orthonormalized function system {9k(q,p)}∞k=0 ∈ L2(R2n) be constructedfrom the linear independent elements of the sequence { fk(q,p)}∞k=0 ∈ L2(R2n) by theGram-Schmidt orthogonalization process [152]. Below we provide strong conver-gence conditions of a series

∑∞

k=0 ak(τ )9k(q,p), where

ak(τ )=

(∞∑i=k

fiτi, 9k

)L2(R2n)

.

Proposition 15.3. Let the generalized Liouville equation (15.2.3) act accordingto (15.2.4); The inequality (15.3.8) and condition cτ/s< 1 are fulfilled. Then theseries

∑∞

k=0 ak(τ )9k(q,p) based upon the an orthonormal system of functions{9k(q,p)}

∞

k=0 ∈ L2(R2n) is strong convergent to the element f (q,p,τ ) ∈ L2(R2n)—thesolution of the Cauchy problem (15.3.4), and∥∥∥∥∥

∞∑k=0

ak(τ )9k(q,p)

∥∥∥∥∥L2(R2n)

≤

∥∥∥∥∥∞∑

k=0

fk(q,p)τk

∥∥∥∥∥L2(R2n)

≤

≤s

(1− τ)(s− cτ)· ‖ f0(q,p)‖L2(R2n),

∞∑k=0

|ak(τ )|2≤

s2

(1− τ)2(s− cτ)2· ‖ f0(q,p)‖

2L2(R2n)

.

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #12


Since∑∞

k=0 |ak(τ )|2 <+∞ and {9k(q,p)}

∞

k=0 ∈ L2(R2n), then according to Riesz-Fisher theorem [2, 147] there exits a unique function f (q,p,τ ) ∈ L2(R2n) such that

ak(τ )= (f ,9k)L2(R2n),

∞∑k=0

|ak(τ )|2= ‖ f‖2L2(R2n)

.

It is well known that ak(τ ) and∑∞

k=0 ak(τ )9k(q,p) are exactly the coeficients andFourier series of the function f (q,p,τ ) ∈ L2(R2n). On the other hand, the Gram matrix

8={9ij} 4= (9i,9j)L2(R2n), i, j= 0,1,2, . . .

of the sequence {9k(q,p)}∞

k=0 ∈ L2(R2n) is bounded and positive definite. Then∑∞

k=0 |ak(τ )|2 <+∞, and according to [270] the series

∑∞

k=0 ak(τ )9k(q,p) stronglyconverges to the function f (q,p,τ ) ∈ L2(R2n).

The constructed above orthonormal system of functions {9k(q,p)}∞k=0 ∈ L2(R2n)

is not complete in L2(R2n). However it is known [152], that every incompleteorthonormal system of functions could be extended to get the complete one by asso-ciating a number of proper functions. In practice to realize such an extension is fairlycomplicated. Since Gram matrix8= {9ij} is bounded and positive definite, the count-able set N = {9k(q,p)}∞k=0 is an orthonormal basis of the subset ¯[9k] [270], i.e., everyelement from ¯[9k] is expanded in a unique strong convergent series. Here ¯[9k] is aclosure of the linear hull [9k].

This approach, to be said—recurrent relation (15.3.4), (15.3.5) also could be usedfor symbolic processing. But as it will be shown later, there exist an other decomposi-tion with a simpler expression. Moreover, the numerical modeling for the small-timetransformation could be done for the known parameter s only. This limitation occuresdue to the convergence conditions stated in the above propositions 15.1, 15.2, and 15.3where the transformation parameter s depends on boundness constant c.

15.3.3 Hermite Polynomial Decomposition

Since the choice of the basis system of functions for the solution decomposition couldbe quite arbitrary, we revised the applicability of the Herimite time–space polynomialdecomposition to the problem (15.2.1). Hence the basic problem statement could bewritten as

∞∑k=0

fk(q,p)H′

k(t)=∞∑

k=0

Lfk(q,p)Hk(t) (15.3.10)

with respect to the definition and the properties of the Poisson bracket.

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #13


Denote Hk(x)—Hermite polynomial defined by recursive expression

Hk+1(x)= 2xHk(x)− 2kHk−1(x), H0 = 1, H1 = 2x;+∞∫−∞

e−x2H2

k (x)dx=√π2kk!.

Proposition 15.4. A solution of the problem (15.2.1) in series (15.3.10) based onorthogonal Hermite polynomial is

f (q,p, t)=∞∑

k=0

fk(q,p)Hk(t) (15.3.11)

fk(q,p)=1

2kLfk−1(q,p)=

1

2kk!Lkf0(q,p). (15.3.12)

Here Lkf = L(Lk−1f ) is an embedded Liouville operator.

Since Hermite polynomials are orthogonal on R we can deal with their orthonormalanalogs

hk(t)=Hk(t)

2k2π

14√

k!,

∫R

e−t2h2k(t)dt = 1.

The corresponding problem solution (15.3.11), (15.3.12) becomes

f (q,p, t)=∞∑

k=0

fk(q,p)hk(t). (15.3.13)

fk(q,p) decomposition coefficients linearly depends form fk(q,p):

fk(q,p)=1√

2kk!Lkf0(q,p). (15.3.14)

To discuss the convergence properties of the (15.3.13) decomposition, we need toremind some basic definitions, see [45, 84, 152] and [269] for example.

Definition 15.2. Let f (x) ∈ L2(V). Then the norm in L2(V) is defined as

‖ f‖ =√(f , f )=

∫V

ω(x)| f (x)|2dx

12

with respect to weight function ω(x) > 0 such that∫

V |ω(x)|2dx<∞.

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #14


Definition 15.3. Let functions f (x),h(x) ∈ L2 and ω(x) > 0—real function are definedin domain V . Then

d( f ,h)= ‖ f − h‖ =

∫V

ω(x)| f (x)− h(x)|2dx

12

is called the distance between functions f and h.

Definition 15.4. Assume that relation

d2(sn,s)=∫V

ω(x)|sn(x)− s(x)|2dx→ 0, n→∞.

is fulfilled for the functions s0(x),s1(x), . . . ,∈ L2(V). Then the sequence {sn}∞

n=0 con-verge to the function s(x) ∈ L2(V) in mean.

First of all we have to note that Hermite polynomials do not belong to L2(R).Nevertheless, assuming V = R2n

×R, we will be able to obtain some useful conver-gence estimations in L2(V).

Theorem 15.3. Let the boundness condition (15.3.8)

‖Lkf0(q,p)‖L2(R2n) ≤ ck‖ f0(q,p)‖L2(R2n), c> 0

holds ∀k = 1,2, . . .. Then∑∞

k=0 ‖ fk(q,p)‖2L2(R2n+1) <∞ and the series (15.3.13) con-

verge in mean with respect to weighting function e−t2 to the solution f (q,p, t) ∈L2(R2n+1) of the problem (15.2.1) and for partial sums sn(q,p, t) of the decompo-sition (15.3.13) holds an estimation

‖sn(q,p, t)− f (q,p, t)‖L2(R2n+1) ≤1√

2n‖ f0(q,p)‖L2(R2n)e

c22 .

Proof. Denote again sn(q,p, t)=∑n

k=0 fk(q,p)hk(t). Hence, according to the defini-tion of the convergence in mean we have to prove that

limn→∞‖sn(q,p, t)− f (q,p, t)‖2L2(R2n+1)

= 0.

Since f (q,p, t) can be written as series (15.3.13), the limit relation becomes

limn→∞‖

∞∑k=n+1

fk(q,p)hk(t)‖2L2(R2n+1)

≤ limn→∞

∞∑k=n+1

‖ fk(q,p)hk(t)‖L2(R2n+1)

2

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #15


‖ fk(q,p)hk(t)‖2L2(R2n+1)

=

∫R2n+1

e−t2| fk(q,p)hk(t)|

2dqdpdt ≤

≤

∫R2n

| fk(q,p)|2dqdp ·

∫R

e−t2h2k(t)dt = ‖ fk(q,p)‖

2L2(R2n)

· 1≤

≤c2k

2kk!‖ f0(q,p)‖

2L2(R2n)

Hence

limn→∞

d2(sn(q,p, t), f (q,p, t))≤ limn→∞

∞∑k=n+1

ck

√2kk!‖ f0(q,p)‖L2(R2n)

2

=

= ‖ f0(q,p)‖2L2(R2n)

limn→∞

∞∑k=n+1

ck

√2kk!

2

According to the Cauchy–Buniakovsky inequality

∞∑k=n+1

ck

√2kk!

2

≤

∞∑k=n+1

1

2k

× ∞∑

k=n+1

c2k

k!

≤≤

1

2n+1

[∞∑

k=0

1

2k

]×

[∞∑

k=0

c2k

k!

]=

1

2n+1· 2 · ec2

=ec2

2n

and

limn→∞

∞∑k=n+1

ck

√2kk!

2

≤ ec2lim

n→∞

1

2n= 0.

15.3.4 Advanced Convergence Results

Using ideas introduced above, we can also assert a generalized result concerning prop-erties for the mean convergence. First of all, assume that ∀k = 1,2, . . .

‖Lkf0(q,p)‖L2(R2n) ≤ ϕ(k) · ‖ f0(q,p)‖L2(R2n), ϕ(k)≥ 0. (15.3.15)

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #16


Function ϕ(k) also could depend upon a number of additional parameters. Hence anew limit relation becomes

limn→∞

d2(sn(q,p, t), f (q,p, t))= limn→∞‖

∞∑k=n+1

fk(q,p)hk(t)‖2L2(R2n+1)

≤

≤ limn→∞

∞∑k=n+1

ϕ(k)√

2kk!‖ f0(q,p)‖L2(R2n)

2

=

= ‖ f0(q,p)‖2L2(R2n)

limn→∞

∞∑k=n+1

ϕ(k)√

2kk!

2

≤

≤ ‖ f0(q,p)‖2L2(R2n)

∞∑k=0

ϕ2(k)

k!× lim

n→∞

∞∑k=n+1

1

2k=

= ‖ f0(q,p)‖2L2(R2n)

∞∑k=0

ϕ2(k)

k!lim

n→∞

1

2n

The last relation reveals the fact that our sequence converge in mean with respect toweighting function e−t2 to the solution f (q,p, t) ∈ L2(R2n+1) of the problem (15.2.1)if series ϕ2(k)/k! is convergent. Then

‖ f0(q,p)‖2L2(R2n)

∞∑k=0

ϕ2(k)

k!lim

n→∞

1

2n≤ Const · ‖ f0(q,p)‖

2L2(R2n)

limn→∞

1

2n= 0

and there holds the following theorem:

Theorem 15.4. Let the boundness condition (15.3.15)

‖Lkf0(q,p)‖L2(R2n) ≤ ϕ(k) · ‖ f0(q,p)‖L2(R2n),ϕ(k)≥ 0

holds ∀k = 1,2, . . . and∑∞

k=0ϕ2(k)

k! = Const�∞. Then∑∞

k=0 ‖ fk(q,p)‖2L2(R2n+1) <

∞ and a series∑∞

k=0 fk(q,p)hk(t) converge in mean with respect to weighting function

e−t2 to the solution f (q,p, t) ∈ L2(R2n+1) of the problem (15.2.1).

Remark 15.3. Comparing the results provided by theorems 15.3 and 15.4 one can see,that from the formal point of view a boundness assumption (15.3.15) is more flexiblethen the initial condition (15.3.8).

Nevertheless, the appropriate choice of ϕ(k) function lead us to different results.For example, using the standard series when k is interpreted as series coefficient

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #17


index only, the application of D’Alambert and Raabe convergence criteria give usthe following results:

ϕ(k+ 1) <√

k+ 1 ϕ(k), ϕ(k+ 1) <√

k ϕ(k).

Since the Raabe test result is more restricted, we can use D’Alambert test resultϕ(k+ 1) <

√k+ 1 ϕ(k). Definitely speaking, the “maximum–maximora” choice here

is ϕ(k)=√(k− 1)!, k = 1,2, . . .which complies with restriction. We can name it sub-

factorial, taking a note that this number becomes bigger then ck for any 0< c�∞starting from some k.

Hence it is important to revise the another one, power series interpretation, assum-ing ϕ(k) equal to αkxk for some unknown parameter x and sequence of constants{αk}

∞

k=0. To discover the convergence of the corresponding power series we need tofind the convergence raduis |x2

|< R (see [1] for example). Namely,

R=1

c, c= lim

n→∞supk>n

(α2

k

k!

) 1k

.

Here we must study two different cases. The first one corresponds to R=∞, thesecond one takes R= Const�∞.

Studying the first case we have two options. One assumes the sequence αk to bebounded, i.e., 0< αk ≤

√M�∞. Then

c= limn→∞

n

√M

n!= 0⇐⇒ R=∞

and

∞∑k=0

ϕ2(k)

k!≤

∞∑k=0

M

k!x2k=M ex2

.

Let x�∞. Then we get a correspondence ck↔Mxk, k = 1,2, . . . between the bound-

ness conditions of the theorems 15.3 and 15.4. They are different, but choosingc=max(Mx,x) (M and x are taken as finite numbers) we satisfy the boundness condi-tion of the first theorem 15.3.

Now assume αk =√

Mdk, 0<M,d�∞. Then

c= limn→∞

d n

√M

n!= 0 ⇐⇒ R=∞.

For the second option we assume x�∞ again and obtain a correspondence ck↔

Mdkx2k, k = 1,2, . . .. Choosing c=max(Mdx2,dx2) we satisfy the boundness condi-tion of the theorem 15.3 again.

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #18


Investigating the second case we have to suppress the factorial element in powerseries coefficient. Let αk =

√Mdkk!, 0<M,d�∞. Then

c= limn→∞

supk>n

(Mdkk!

k!

) 1k

= limn→∞

d n√

M = d.

According to the definition of the power series convergence radius, assume 0< x2 <

1/d. Then we get the following result:

ck↔M

dkk!

dk⇐⇒ ck

↔Mk!, k = 1,2, . . .

with corresponding factorial (extended) boundness condition for mean convergence:

‖Lkf0(q,p)‖L2(R2n) ≤Mk! · ‖ f0(q,p)‖L2(R2n), 0<M�∞. (15.3.16)

Since boundness parameters c and M are finite numbers, and factorial growsasymptotically faster then a number’s degree, then we can combine theorems 15.3,15.4 in the following manner:

Corollary 15.1. Let the boundness condition

‖Lkf0(q,p)‖L2(R2n) ≤max(M1ck, M2k!) · ‖ f0(q,p)‖L2(R2n),

0< c;M1,M2�∞

holds ∀k=1,2, . . . Then∑∞

k=0 ‖ fk(q,p)‖2L2(R2n+1)<∞ and a series

∑∞

k=0 fk(q,p)hk(t)

converge in mean with respect to weighting function e−t2 to the solution f (q,p, t) ∈L2(R2n+1) of the problem (15.2.1).

It is evident that boundness condition appeared in corollary 15.1 is much moreflexible and extends the applicability of the proposed solution technique.

15.3.5 Partial Conclusions

The advantages of the used polynomial decompositions are the relative simplicity ofcoefficient recurrence relations and their convergence properties both in the infiniteinitial and finite transformed solution time domains. Assuming Liouville operator L tobe bounded (15.3.8), we can choose a proper small time space transformation param-eter s:τ = 1− e−st to guarantee strong or/and weak asymptotic series convergence tothe solution of the problem (15.3.4) in R2n.

By contrast we have a fast growing computational complexity of evaluating embed-ded Liouville opearator even for two and three dimensional generalized vectors q,p.Hence all this techinques mainly could be used for qualitative solution analysis. Nev-ertheless, once we have some analytic truncated series approximation, we are free tochoose any kind of trayectories and impulse vectors dependencies for numerical mod-eling and/or visualization.

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #19


15.4 Eigen Expansion of Generalized Liouville Operator

The other classical approach to solve Cauchy problem (15.3.4) assumes that expansionof the solution f (q,p,τ ) over the system of eigenfunctions {gk(q,p)}∞k=0 to be found.Here we needs the generalized Liouville operator (15.2.3) defined on the test functionsC∞0 (R

2n). Thus, translating them into complex Hilbert space L2(R2n) we are able tofullfill our task.

Without loss of generality let us assume {gk(q,p)}∞k=0 to be an orthonormalsequence of eigenfunctions of operator (15.2.3), complete in R(L)⊂ L2(R2n); {λk}

∞

k=0is corresponding sequence of eigenvalues;∫

R2n

gk(q,p)gn(q,p)dqdp= δkn

Lgk(q,p)= λkgk(q,p), gk(q,p) ∈ D(L)= C∞0 (R2n).

The solution of the Cauchy problem (15.3.4) we will seek in the form of asymptoticexpansion

f (q,p,τ )=∞∑

k=0

vk(τ )gk(q,p), vk(τ )=

∫R2n

f (q,p,τ )gk(q,p)dqdp. (15.4.1)

Let τ = 0. Then (15.4.1) becomes

f (q,p,0)= f0(q,p)=∞∑

k=0

vk(0)gk(q,p).

To find the functions vk(τ ) we pose the Cauchy problem

(1− τ)∂

∂τ{vk(τ )gk(q,p)} =

1

sL{vk(τ )gk(q,p)},

vk(τ )|τ=0 = vk(0)4= v0

k

whose solution has the form

vk(τ )= v0k · (1− τ)

−λk/s. (15.4.2)

Formula (15.4.2) could be derived directly from (15.4.1). In fact, differentiating(15.4.1), we get

d

dτvk(τ )=

1

s(1− τ)· (Lf (q,p,τ ),gk(q,p))L2(R2n).

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #20


Since there holds an identity

(Lf (q,p,τ ),gk(q,p))L2(R2n) = vk(τ ) · (Lgk(q,p),gk(q,p))L2(R2n),

then the relation

d

dτvk(τ )=

λk

s(1− τ)· vk(τ )

is fulfilled. Taking vk(τ )|τ=0 = v0k , it immediately follows (15.4.2). Function f (q,p,τ )

defined by a series (15.4.1) with a coefficients defined by expression (15.4.2) is thesolution of the Cauchy problem (15.3.4).

A series∑∞

k=0(f ,gk)L2(R2n) · gk will be convergent independently of the termsorder. Moreover, the sum is equal to f (q,p,τ ) for f (q,p,τ ) ∈ [L], where L={gk(q,p)}∞k=0.

Therefore a series (15.4.1) can be differentiated termwise

∂

∂τf (q,p,τ )=

∞∑k=0

vk(τ )gk(q,p), vk(τ )=d

dτvk(τ ).

Multiplying both sides of previous equality by (1− τ) and taking into account that

(1− τ)gk(q,p) · vk(τ )=1

svk(τ )Lgk(q,p)

we obtain

(1− τ) ·∂

∂τf (q,p,τ )=

1

s

∞∑k=0

vk(τ )Lgk(q,p)=

1

sL

{∞∑

k=0

vk(τ )gk(q,p)

}=

1

sLf (q,p,τ ).

Here f (q,p,τ )|τ=0 = f0(q,p). Thus we derived the formula

f (q,p,τ )=∞∑

k=0

v0k · (1− τ)

−λk/s · gk(q,p) (15.4.3)

representing the solution expansion of the generalized Liouville equation with respectto complete orthonormal system of eigenfunctions {gk(q,p)}∞k=0 of operator L in R(L).

Making the backward time scale substitution we transform (15.4.3) into

f (q,p, t)=∞∑

k=0

v0k · e

λkt· gk(q,p).

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #21


Now we have had to pay attention for evaluating eigenfunctions gk(q,p) and eigen-values λk of the operator L. To this end we take an element gi(q,p) and consider theexpansion with respect to orthonormal function system

gi(q,p)= r0iψ0(q,p)+ r1iψ1(q,p)+ ·· ·+ rijψj(q,p)+ ·· · (15.4.4)

where rji = (gi,ψj)L2(R2n). The equality (15.4.4) holds for gi(q,p) ∈ [ψk]. Since a

series on the right of (15.4.4) is convergent independently of the terms order, thereholds an equility

Lgi(q,p)= r0iLψ0(q,p)+ r1iLψ1(q,p)+ ·· ·+ rjiLψj(q,p)+ ·· ·

or

λigi(q,p)= r0iLψ0(q,p)+ r1iLψ1(q,p)+ ·· ·+ rjiLψj(q,p)+ ·· · . (15.4.5)

Denote

xik = (gi,ψk)L2(R2n), aik = (Lψi,ψk)L2(R2n). (15.4.6)

Scalar product of (15.4.5) and function ψk(q,p) could be written

λixik = r0ia0k+ r1ia1k+ ·· ·+ rjiajk+ ·· ·

or

λixik = xi0a0k+ xi1a1k+ ·· ·+ xijajk+ ·· ·

in notations (15.4.6). The last relation could be expressed in the standard matrix form

λiXi = AXi. (15.4.7)

Thus expansion coefficients rji4= (gi,ψj)L2(R2n)

=xij of the function gi(q,p) in aseries (15.4.4) are set by a linear algebraic problem on eigenvalues (15.4.7) for infinitematrix A.

Taking some given number of elements in truncated series (15.4.1) the problem(15.4.7) could be solved for certain finite matrix A. Here λi and gi(q,p) are complexin general, matrix A—non simmetrical one. But under certain suppositions an operatoriL, i2 =−1 will be simmetrical and even selfajoint. In this particular case λi are realand gi(q,p) are imaginal. Assuming this, one can derive an efficient algorithms for

solving (15.4.7) for a coefficients ajk4= (Lψj,ψk)L2(R2n)

= akj. Using natural notationsfor the Gram-Schmidt orthonormalization process we get

ψi(q,p)= α0f0(q,p)+α1f1(q,p)+ ·· ·+αifi(q,p),

fi(q,p)= γ0ψ0(q,p)+ γ1ψ1(q,p)+ ·· ·+ γiψi(q,p).

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Therefore the following chain of equalities holds

Lψi(q,p)= α0Lf0(q,p)+α1f1(q,p)+ ·· ·+αiLfi(q,p)=

= ξ1f1(q,p)+ ξ2f2(q,p)+ ·· ·+ ξifi(q,p)+ ξi+1fi+1(q,p)=

= β0ψ0(q,p)+β1ψ1(q,p)+ ·· ·+βiψi(q,p)+βi+1ψi+1(q,p),

and if j> i+ 1, then (Lψi,ψj)L2(R2n) = (ψi,Lψj)L2(R2n) = 0. Hence matrix A is a bandmatrix, or exactly speaking, it has a three diagonal form

A=

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

a1 b1 0b1 a2 b2

b2 a3 b3

b3 a4. . .

. . .. . .

. . .

0. . .

. . .

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥. (15.4.8)

The eigenvalue problem for the banded matrices a well studied and a proper algorithmscould be found in hundreds of books and research articles. The standard numericallystable approach like QR-shift method could be directly applied to the matrix A of thegiven structure. Fixing the number of series (15.4.3) terms we solve problem (15.4.7)for matrix A (15.4.8) of some finite dimension. For larger dimensions—the case ofthe special interest in practice—the computational error grows and elements ψi(q,p)loose their orthogonal property in numerics.

Making a short resume, the eigen decomposition technique hardly could be recom-mended for general practical applications especially for complicated analitical initialdistribution functions f0(q,p) since they generate linear systems of complex valuednonsymmetric matrices turning the truncated solution to be numerical one. The numer-ical stability characteristics are uncertain. Also we have a tripled (at least) computa-tional requirements.

15.5 Hermitian Function Expansion

Since the standard Hermite polynomial provides only a convergence in mean (seetheorem 15.1) and Hk(t), hk(t) do not belong to L2(R), we are interested in obtainingsome other expansions over a certain set of functions {uk}

∞

k=0 ∈ L2(R). Such functionsbased on Hermite orthogonal polynomials are usually called Hermitian [269], or asso-ciated Hermite functions, see Chapter 22, “Orthogonal Polynomials” [1]. Formallythey are constructed as a parametric family of orthonormal functions with additional

DULOV 19-ch15-261-286-9780123877796 2011/5/24 23:08 #23


useful properties

un(t)=√

a√πn!2n

Hn(at)e−at22 , 0< a<∞ (15.5.1)∫

R

un(t)um(t)dt = δn,m

∫R

un(t)u′m(t)dt = a

√n+ 1

2δn,n+1− a

√n

2δn,n−1. (15.5.2)

Here δn,m is Kronecker’s delta.Hence, using the standard expansion of system (15.2.1) in time domain (15.3.10)

with respect to definition (15.5.1), (15.5.2) we will obtain

fn+1(q,p)=1

a

√2

n+ 1Lfn(q,p)+

√n

n+ 1fn−1(q,p), n= 0,1, . . . (15.5.3)

f−1(q,p)≡ 0.

Denote ν =

√2

ato simplify the further expressions. Then one can prove by direct

computations that

f2k(q,p)=1

√(2k)!

k∑s=0

λ(2k)s ν2sL2sf0(q,p), k = 1,2, . . . (15.5.4)

f2k+1(q,p)=1

√(2k+ 1)!

k∑s=0

λ(2k+1)s ν2s+1L2s+1f0(q,p), k = 1,2, . . . (15.5.5)

where λ(n)s ∈ N is a coefficient in a emdedded Liouville operator power series decom-position.

Combining formal expansions (15.5.4), (15.5.5) with recurrent coefficient formula(15.5.3) and recalling that (2k− 1)!!= (2k− 1) · (2k− 3) · . . . · 3 · 1 one can obtaindetailed coefficient dependencies for even elements:

λ(2k)k ≡ 1, k = 1,2 . . .

λ(2k)0 ≡ (2k− 1)!!, k = 1,2 . . .

λ(2k)k−1 ≡ (2k− 1)k, k = 2,3 . . .

(15.5.6)

and for the odd ones:

λ(2k+1)k ≡ 1, k = 1,2 . . .

λ(2k+1)0 ≡ (2k+ 1)!!, k = 1,2 . . .

λ(2k+1)k−1 ≡ (2k+ 1)k, k = 2,3 . . .

(15.5.7)

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Since {uk(t)}∞

k=0 ∈ L2(R) is orthonormal, then using partial coefficient value analy-sis we can revise the applicability of Hermitian function expansion in sense of Riesz–Fischer theorem.

If∞∑

k=0

‖ fk(q,p)‖2L2(R2n)

<∞, then∞∑

k=0

fk( p,q)uk(t)→ f (q,p, t)

to the unique function f (q,p, t) ∈ L2(R2n+1).Assume that Liouville operator boundness condition (15.3.8) holds:

‖Lkf0(q,p)‖L2(R2n) ≤ ck· ‖ f0(q,p)‖L2(R2n).

Then, due to a difference in odd/even coefficients we obtain

∞∑k=0

‖ fk(q,p)‖2L2(R2n)

=

∞∑k=0

‖ f2k(q,p)‖2L2(R2n)

+

∞∑k=0

‖ f2k+1(q,p)‖2L2(R2n)

=

=

∞∑k=0

1

(2k)!

∥∥∥∥∥k∑

s=0

λ(2k)s ν2sL2sf0(q,p)

∥∥∥∥∥2

L2(R2n)

+

+

∞∑k=0

1

(2k+ 1)!

∥∥∥∥∥k∑

s=0

λ(2k+1)s ν2s+1L2s+1f0(q,p)

∥∥∥∥∥2

L2(R2n)

≤

≤ ‖ f0(q,p)‖2L2(R2n)

∞∑k=0

[∑k

s=0λ(2k)s (cν)2s

]2

(2k)!+

[∑ks=0λ

(2k+1)s (cν)2s+1

]2

(2k+ 1)!

.∞∑

k=0

‖ fk(q,p)‖2L2(R2n)

≤ ‖ f0(q,p)‖2L2(R2n)

× (15.5.8)

×

∞∑k=0

(2k+ 1)[∑k

s=0λ(2k)s (cν)2s

]2+ (cν)2

[∑ks=0λ

(2k+1)s (cν)2s

]2

(2k+ 1)!.

One can see that with respect to k→∞ the odd and even internal sums in theabove expression could be treated as a partial sums of an ordinary power series.Here the major coefficient is 1 according to (15.5.6), (15.5.7). Conversely, a conver-gence radius R (see [1]) R≡ 1. This result provides us with a “partial” convergencecondition

cν < 1 ⇒ a> c√

2. (15.5.9)

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Now, assume c> 0, a> 0 to be some constants and conditions (15.3.8), (15.5.9)are fulfilled. Denote

ξ(c,a)=max

supk→∞

k∑s=0

λ(2k)s

(c√

2

a

)2s

, supk→∞

k∑s=0

λ(2k+1)s

(c√

2

a

)2s<∞.

Using an introduced notation ξ(c,a), the estimation (15.5.8) becomes

∞∑k=0

‖ fk(q,p)‖2L2(R2n)

≤ ‖ f0(q,p)‖2L2(R2n)

ξ2(c,a)∞∑

k=0

(2k+ 1)+ (cν)2

(2k+ 1)!=

= ‖ f0(q,p)‖2L2(R2n)

ξ2(c,a)(

cosh(1)+ (cν)2sinh(1))<∞

(15.5.10)

This proves the following result:

Theorem 15.5. Suppose Liouville operator (15.2.3) to be bounded (15.3.8)and parameter a for the associated Hermite functions (15.5.1) a> c

√2. Then∑

∞

k=0 ‖ fk(q,p)‖2L2(R2n)<∞ and according to Riesz–Fisher theorem the series are

convergent

∞∑k=0

fk(q,p)uk(t)→ f (q,p, t), f (q,p, t) ∈ L2(R2n+1)

to the unique function.

Comparing results stated in propositions 15.1, 15.2, 15.3, and theorems 15.3, 15.5there arise some limitations for it’s analytical/numerical applications as expansionseries:

1. The formal boundness assumption (15.3.8) is the sufficient and necessory condition to ensurethe convergence of the expansion series for small–time space method or Hermite–baseddecompositions;

2. To ensure the applicability of Riesz–Fisher theorem, one should exactly know the value ofthe bounding constant c. This value is needed for evaluating the correct small–time parame-ter s or Hermite associated function parameter a;

3. Supposing Liuoville operator boundness constant c to be finite unknown number, we canguarantie only the convergence in mean for standard Hermite polynomial decomposition.

15.6 Another Application Example for HermitePolynomial Decomposition

A novel approach proposing decomposition of the distribution function fj(t,x,v),index letter j= i,e for ions and electrons respectively was proposed and successfully

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applied by Japanese scientists A. Suzuki and T. Shigeyama [266]. They took Vlasov-Maxwell system of equations in the R×R3

×R3 phase space as a model of collision-less plasmas. Assuming plasma to be stationary and homogeneous in one directionwe have physical variables independent from time t and selected direction, for exam-ple z. Hence we can introduce Hermitian decomposition depending on relation vz/vj,j= i,e. And once again the orthogonal property of Hermite polynomials produced arecurrent relation for decomposition coefficients. Studying several different types ofequilibrium conditions they were able to derive analytical solutions as well as for non-Maxwellian equilibrium making a deep insight into the properties of self-consistentplasma configurations. Since this paper was an initial step it does not provide any kindof general convergence conditions or theorems. But in our case the most important isthe applicability of the Hermite-type decompositions for a wide range of fundamentalequations describing the building of modern physics.

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Glossary of Terms and Symbols

1D one-(single) dimensional3D three-dimensionalBEq Branching equationBVP Boundary-value problemDM-BE Discrete models of Boltzmann equationKdV Korteweg-de VriesKE Kinetic equationLGA Lattice gases automataMC Monte-CarloODE Ordinary differential equationQH Quantum HamiltonianRSS Rudykh-Sidorov-SinitsynVM Vlasov-MaxwellVP Vlasov-Poisson

DULOV bibliography-9780123877796 2011/5/26 13:05 Page 289 #1

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