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Some Applications of an Energy Method in Collisional Kinetic Theory
by
Robert Mills Strain III
B. A., New York University, 2000
Sc. M., Brown University, 2002
A Dissertation submitted in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy
in the Division of Applied Mathematics at Brown University
Providence, Rhode Island
May 2005
c� Copyright 2005 by Robert Mills Strain III
Contents
Chapter 1. Introduction 1
1.1. Precise statements of the results 2
Chapter 2. Stability of the Relativistic Maxwellian in a Collisional Plasma 8
2.1. Collisional Plasma 8
2.2. Main Results 13
2.3. The Relativistic Landau Operator 21
2.4. Local Solutions 55
2.5. Positivity of the Linearized Landau Operator 65
2.6. Global Solutions 75
Chapter 3. Almost Exponential Decay near Maxwellian 80
3.1. Introduction 80
3.2. Vlasov-Maxwell-Boltzmann 82
3.3. Relativistic Landau-Maxwell System 86
3.4. Soft Potentials 87
3.5. The Classical Landau Equation 93
Chapter 4. Exponential Decay for Soft Potentials near Maxwellian 95
4.1. Introduction 95
4.2. Introduction 95
4.3. Boltzmann Estimates 103
4.4. Landau Estimates 125
4.5. Energy Estimate and Global Existence 150
4.6. Proof of Exponential Decay 154
Chapter 5. On the Relativistic Boltzmann Equation 156
v
5.1. The Relativistic Boltzmann Equation 157
5.2. Examples of relativistic Boltzmann cross-sections 162
5.3. Lorentz Transformations 164
5.4. Center of Momentum Reduction of the Collision Integrals 173
5.5. Glassey-Strauss Reduction of the Collision Integrals 175
5.6. Hilbert-Schmidt form 179
5.7. Relativistic Vlasov-Maxwell-Boltzmann equation 187
Appendix A. Grad’s Reduction of the Linear Boltzmann Collision Operator 189
Bibliography 196
vi
Abstract of “Some Applications of an Energy Method in Collisional Kinetic Theory”
by Robert Mills Strain III, Ph.D., Brown University, May 2005
The collisional Kinetic Equations we study are all of the form
@tF + v ·rxF + V (t, x) ·rvF = Q(F, F ).
Here F = F (t, x, v) is a probabilistic density function (of time t � 0, space x 2 ⌦ and
velocity v 2 R3) for a particle taken chosen randomly from a gas or plasma. V (t, x) is
a field term which usually represents Maxwell’s theory of electricity and magnetism,
sometimes this term is neglected. Q(F, F ) is the collision operator which models the
interaction between colliding particles. We consider both the Boltzmann and Landau
collision operators.
We prove existence, uniqueness and regularity of close to equilibrium solutions to
the relativistic Landau-Maxwell system in the first part of this thesis. Our main tool
is an energy method.
In the second part, we prove arbitrarily high polynomial time decay rates to
equilibrium for four kinetic equations. These are cuto↵ soft potential Boltzmann and
Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic
Landau-Maxwell system. The main technique used here is interpolation.
In the third part, we prove exponential decay for the cuto↵ soft potential Boltz-
mann and Landau equations. The main point here is to show that exponential decay
of the initial data is propagated by a solution.
In the fourth and final part of this thesis, we write down a few important calcula-
tions in the relativistic Boltzmann theory which are scattered around the literature.
We also calculate a few Lorentz transformations which maybe useful in relativistic
transport theory. We use these calculations to comment about extending the results
in this thesis to the relativistic Boltzmann equation.
CHAPTER 1
Introduction
In this thesis, we study some mathematical properties of a class of nonlinear
partial di↵erential equations called “kinetic equations” which arise out of the Kinetic
Theory of Gases and Plasmas. We are primarily concerned with stability theorems,
e.g. results on existence, uniqueness and decay rates of solutions which are initially
close to steady state.
1.0.1. Brief Introduction to Kinetic Theory. A rarefied gas is a large collec-
tion of electrically neutral particles (around 1023) in which collisions form a small part
of each particles lifetime. A plasma is a large collection of fast-moving charged parti-
cles. The large number of particles makes it di�cult to attempt a physical description
of a gas or plasma using Classical Mechanics.
Boltzmann (1872) instead decided to posit the existence of a probability density
function F (t, x, v) which roughly speaking tells you the average number of particles
a time t, position x and velocity v. He proposed a partial di↵erential equation which
governs the time evolution of F . Boltzmann’s Equation is the foundation of kinetic
theory [11, 12, 15, 30, 50, 67, 68]. The density function F (t, x, v) contains a great deal
of information, it can be a practical tool for determining detailed properties of dilute
gases and plasmas as well as calculating many important physical quantities.
Kinetic Theory has been among the most active areas in the mathematical study of
nonlinear partial di↵erential equations over the last few decades–spectacular progress
has been made yet still more is expected. The main goal of this research is to study
the existence, uniqueness, regularity and asymptotic properties of solutions F . And
the major result thus far is the global renormalized weak solutions of Diperna & Lions
[23] to the Boltzmann equation with large initial data.
1
Recently, Yan Guo has developed a flexible nonlinear energy method [39] used
to construct global in time smooth solutions to the “hard sphere” Vlasov-Maxwell-
Boltzmann system for initial data close to Maxwellian equilibrium. This system is
a generalization of the Boltzmann equation which models two species of particles
(electrons and ions) interacting and includes electromagnetic e↵ects. It is a system
of ten partial di↵erential equations, two coupled Boltzmann equations that represent
electrons and ions interacting also coupled with Maxwell’s equations for electricity
and magnetism. See [35–40] for more applications of such a method. There is also
related work near vacuum [41]. In this Ph.D thesis, we apply this energy method a
few important problems in kinetic theory.
1.1. Precise statements of the results
In this doctoral thesis, I develop new techniques to study existence, uniqueness,
regularity and asymptotic convergence rates of several nonlinear partial di↵erential
equations from collisional Kinetic Theory. In Section 1.1.1, we discuss the relativistic
Landau-Maxwell system. In Section 1.1.2, we discuss the asymptotic decay of four
kinetic equations. And in Section 1.1.3, we discuss exponential decay for soft potential
Boltzmann and Landau equations.
1.1.1. Relativistic Landau-Maxwell System. Dilute hot plasmas appear com-
monly in important physical problems such as Nuclear fusion and Tokamaks. The
relativistic Landau-Maxwell system [50] is the most fundamental and complete model
for describing the dynamics of a dilute collisional plasma in which particles interact
through binary Coulombic collisions and through their self-consistent electromagnetic
field. It is widely accepted as the most complete model for describing the dynamics
of a dilute collisional fully ionized plasma.
Yan Guo and I proved existence of global in time classical solutions [62] with
initial data near the relativistic Maxwellian. To the best of our knowledge, this is the
first existence proof for the relativistic Landau-Maxwell system. In 2000, Lemou [47]
studied the linearized relativistic Landau equation with no electromagnetic field. In
2
the non-relativistic situation there are a few results for classical solutions to kinetic
equations with the Landau collision operator [17, 38,72].
We considered the following coupled relativistic Landau-Maxwell system (two
equations):
@tF± +p
p0
·rxF± ±✓
E +p
p0
⇥B
◆
·rpF± = C(F±, F±) + C(F±, F⌥)
Here F±(t, x, p) � 0 are the spatially periodic number density functions for ions (+)
and electrons (�) at time t � 0, position x = (x1
, x2
, x3
) 2 T3 = [�⇡, ⇡]3 and
momentum p = (p1
, p2
, p3
) 2 R3. The energy of a particle is given by p0
=p
1 + |p|2.
The electromagnetic field, E(t, x) and B(t, x), is coupled with F±(t, x, p) through the
celebrated Maxwell system:
@tE �rx ⇥B = �4⇡
Z
R3
p
p0
{F+
� F�} dp, @tB +rx ⇥ E = 0,
with constraints rx · B = 0, rx · E = 4⇡R
R3 {F+
� F�} dp and spatially periodic
initial conditions.
The relativistic Landau collision operator is defined by
C(G, H) ⌘ rp ·Z
R3
�(p, q) {rpG(t, x, p)H(t, x, q)�G(t, x, p)rqH(t, x, q)} dq,
where the 3⇥ 3 matrix �(p, q) is a complicated expression [47,50,62].
Steady state solutions are [F±(t, x, p), E(t, x), B(t, x)] = [e�p0 , 0, B], where B is a
constant which is determined initially. We consider the standard perturbation around
the relativistic Maxwellian F±(t, x, p) = e�p0 + e�12p0f±(t, x, p). Define the high order
energy norm for a solution to the relativistic Landau-Maxwell system as
EN(t) ⌘ 1
2|||[f
+
, f�]|||2N(t) + |||[E, B]|||2N(t) +
Z t
0
|||[f+
, f�]|||2�,N(s)ds,
where |||·|||N represents an L2 sobolev norm in (x, p) (or just in x in the case of [E, B])
but also includes t derivatives. ||| · |||N contains (t, x, p) derivatives of all orders N .
The dissipation ||| · |||�,N represents the same norm, but with a momentum weight
and one extra p-derivative. At t = 0 the temporal derivatives are defined naturally
through the equations.
3
Theorem 1.1. [62]. Fix N � 4. Let F0,±(x, p) = e�p0 + e�
12p0f
0,±(x, p) � 0.
Assume that initially [F0,±, E
0,B0
] has the same mass, total momentum and total
energy as the steady state [e�p0 , 0, B].
9C > 0, ✏ > 0 such that if EN(0) ✏ then there exists a unique global in time solu-
tion [F±(t, x, p), E(t, x), B(t, x)] to the relativistic Landau-Maxwell system. Moreover,
F±(t, x, p) = e�p0 + e�12p0f±(t, x, p) � 0
and sup0s1 EN(s) CEN(0).
The proof of this theorem uses the energy method from [39]. However, severe
new mathematical di�culties were present in terms of the complexity of the relativis-
tic Landau Collision kernel �(p, q) and the momentum derivatives in the collision
operator C. One important question left open in this result and in the case of the
Vlasov-Maxwell-Boltzmann system [39] is the question of a precise time decay rate
to equilibrium.
1.1.2. Almost Exponential Decay Near Maxwellian. The key di�culty in
obtaining time decay for the relativistic Landau-Maxwell system is contained in the
following di↵erential inequality (�N > 0):
dyN
dt(t) +
�N2|||[f
+
, f�]|||2�,N(t) 0,
where the instant energy, yN(t), is equivalent to
1
2|||[f
+
, f�]|||2N(t) + |||[E, B � B]|||2N(t).
The di�culty lies in the fact that the instant energy functional at each time is stronger
than the dissipation rate |||[f+
, f�]|||2�,N(t):
|||[f+
, f�]|||2N(t) + |||[E, B � B]|||2N�1
(t) C|||[f+
, f�]|||2�,N(t).
No estimates for the N-th order derivatives of E and B are available. It is therefore
di�cult to imagine being able to use a Gronwall type of argument to get the time
decay rate. Our main observation is that a family of energy estimates have been
4
derived in [62]. The instant energy is stronger for fixed N , but it is possible to be
bounded by a fractional power of the dissipation rate via simple interpolations with
stronger energy norms. Only algebraic decay is possible due to such interpolations,
but more regular initial data grants faster decay.
Theorem 1.2. [60]. Fix N � 4, k � 1 and choose initial data [F0,±, E
0
, B0
] which
satisfies the assumptions of the previous Theorem for both N and N + k. Then there
exists CN,k > 0 such that
|||[f+
, f�]|||2N(t) + |||[E, B � B]|||2N(t) CN,kEN+k(0)
✓
1 +t
k
◆�k
.
And the same decay holds for the Vlasov-Maxwell-Boltzmann system under similar
conditions.
On a related note, Desvillettes and Villani have recently undertaken an impressive
program to study the time-decay rate to Maxwellian of large data solutions to soft
potential Boltzmann type equations. Even though their assumptions in general are
a priori and impossible to verify at the present time, their method does lead to an
almost exponential decay rate for the soft potential Boltzmann and Landau equations
for solutions close to a Maxwellian. This surprising new decay result relies crucially
on recent energy estimates in [37, 38] as well as other extensive and delicate work
[18–21, 53, 64, 69]. To obtain time decay for these models with very ‘weak’ collision
e↵ects has been a very challenging open problem even for solutions near a Maxwellian,
therefore it is of great interest to find simpler proofs. Using interpolations with higher
velocity weights, Yan Guo and I gave a much simpler proof of almost exponential
decay for the Boltzmann and Landau equations.
The Boltzmann equation is written as
@tF + v ·rxF = Q(F, F ), F (0, x, v) = F0
(x, v).
Above F (t, x, v) is the spatially periodic density function for particles at time t � 0,
position x = (x1
, x2
, x3
) 2 T3 and velocity v = (v1
, v2
, v3
) 2 R3. The collision operator
5
is
Q(F, F ) ⌘Z
R3
du
Z
S2
d!B(✓)|u� v|�{F (t, x, u0)F (t, x, v0)� F (t, x, u)F (t, x, v)}.
Here ✓ = ✓(u, v,!) is the scattering angle and [u0, v0] = [u0(u, v,!), v0(u, v,!)] are
the post-collisional velocities. The exponent in the collision operator is �3 < � < 0
(soft potentials) and we assume B(✓) satisfies the Grad angular cuto↵ assumption:
0 < B(✓) C| cos ✓|. The Landau Equation is obtained in the so-called “grazing
collision” limit of the Boltzmann equation [50].
We write the perturbation from Maxwellian, µ = e�|v|2, as
F (t, x, v) = µ(v) +p
µ(v)f(t, x, v).
Then define the high order energy norm
(1.1) E`(t) ⌘1
2|||f |||2`(t) +
Z t
0
|||f |||2⌫,`(s)ds,
where |||f |||2`(t) is an instantaneous norm in time and an energy norm in the space and
velocity variables which includes derivatives in all variables (t, x, v) and of all orders
N where N � 8. Here ` denotes a polynomial velocity weight like (1 + |v|2)��`/2.
At t = 0 we define the temporal derivatives naturally through the equation. ||| · |||⌫,`
is the same norm as ||| · |||`, but with another dissipation weight.
Theorem 1.3. [60]. Fix �3 < � < 0; fix integers ` � 0 and k > 0. Let
F0
(x, v) = µ +p
µf0
(x, v) � 0.
Assume the initial data F0
(x, v) has the same mass, momentum and energy as the
steady state µ.
9C` > 0, ✏` > 0 such that if E`(0) < ✏` then there exists a unique global in time
solution to the soft potential Boltzmann equation
F (t, x, v) = µ + µ1/2f(t, x, v) � 0
with sup0s1 E`(f(s)) C`E`(0).
6
Moreover, if E`+k(0) < ✏`+k, then the unique global solution satisfies
|||f |||2`(t) C`,kE`+k(0)
✓
1 +t
k
◆�k
.
The same result holds for the Landau Equation.
We remark that the existence for both the Boltzmann and Landau case were
proven in [37,38] at ` = 0.
1.1.3. Exponential Decay Near Maxwellian. These “almost exponential”
decay results suggest strongly that under some conditions exponential decay can
be achieved. Indeed, in the case of the Boltzmann equation with a soft potential
Caflisch [8, 9] got decay of the form exp(��tp) for some � > 0 and 0 < p < 1 but
only for �1 < � < 0. Caflisch uses an exponential weight of the for eq|v|2 in his
norms for 0 < q < 1
4
. Yan Guo and I extend Caflisch’s result to the soft potential
Boltzmann equation for the full range �3 < � < 0 and to the Landau equation using
an exponential weight like eq|v|# for 0 # 2.
7
CHAPTER 2
Stability of the Relativistic Maxwellian in a Collisional
Plasma
Abstract. The relativistic Landau-Maxwell system is the most fundamental and
complete model for describing the dynamics of a dilute collisional plasma in which
particles interact through Coulombic collisions and through their self-consistent elec-
tromagnetic field. We construct the first global in time classical solutions. Our
solutions are constructed in a periodic box and near the relativistic Maxwellian, the
Juttner solution. This result has already appeared in a modified form as [62].
2.1. Collisional Plasma
A dilute hot plasma is a collection of fast moving charged particles [42]. Such
plasmas appear commonly in such important physical problems as in Nuclear fusion
and Tokamaks. Landau, in 1936, introduced the kinetic equation used to model a
dilute plasma in which particles interact through binary Coulombic collisions. Landau
did not, however, incorporate Einstein’s theory of special relativity into his model.
When particle velocities are close to the speed of light, denoted by c, relativistic e↵ects
become important. The relativistic version of Landau’s equation was proposed by
Budker and Beliaev in 1956 [3–5]. It is widely accepted as the most complete model
for describing the dynamics of a dilute collisional fully ionized plasma.
The relativistic Landau-Maxwell system is given by
@tF+
+ cp
p+
0
·rxF+
+ e+
✓
E +p
p+
0
⇥B
◆
·rpF+
= C(F+
, F+
) + C(F+
, F�)
@tF� + cp
p�0
·rxF� � e�
✓
E +p
p�0
⇥B
◆
·rpF� = C(F�, F�) + C(F�, F+
)
8
with initial condition F±(0, x, p) = F0,±(x, p). Here F±(t, x, p) � 0 are the spatially
periodic number density functions for ions (+) and electrons (�), at time t � 0,
position x = (x1
, x2
, x3
) 2 T3 ⌘ [�⇡, ⇡]3 and momentum p = (p1
, p2
, p3
) 2 R3. The
constants ±e± and m± are the magnitude of the particles charges and rest masses
respectively. The energy of a particle is given by p±0
=p
(m±c)2 + |p|2.
The l.h.s. of the relativistic Landau-Maxwell system models the transport of
the particle density functions and the r.h.s. models the e↵ect of collisions between
particles on the transport. The heuristic derivation of this equation is
total derivative along particle trajectories = rate of change due to collisions,
where the total derivative of F± is given by Newton’s laws
x = the relativistic velocity =p
p
m± + |p|2/c,
p = the Lorentzian force = ±e±
✓
E +p
p±0
⇥B
◆
.
The collision between particles is modelled by the relativistic Landau collision oper-
ator C in (2.1) and [3, 4, 50] (sometimes called the relativistic Fokker-Plank-Landau
collision operator).
To completely describe a dilute plasma, the electromagnetic field E(t, x) and
B(t, x) is generated by the plasma, coupled with F±(t, x, p) through the celebrated
Maxwell system:
@tE � crx ⇥B = �4⇡J = �4⇡
Z
R3
⇢
e+
p
p+
0
F+
� e�p
p�0
F�
�
dp,
@tB + crx ⇥ E = 0,
with constraints
rx ·B = 0, rx · E = 4⇡⇢ = 4⇡
Z
R3
{e+
F+
� e�F�} dp,
and initial conditions E(0, x) = E0
(x) and B(0, x) = B0
(x). The charge density and
current density due to all particles are denoted ⇢ and J respectively.
9
We define relativistic four vectors as P+
= (p+
0
, p) = (p+
0
, p1
, p2
, p3
) and Q� =
(q�0
, q). Let g+
(p), h�(p) be two number density functions for two types of particles,
then the Landau collision operator is defined by
C(g+
, h�)(p) ⌘ rp ·Z
R3
�(P+
, Q�) {rpg+
(p)h�(q)� g+
(p)rqh�(q)} dq.(2.1)
The ordering of the +,� in the kernel �(P+
, Q�) corresponds to the order of the
functions in argument of the collision operator C(g+
, h�)(p). The collision kernel is
given by the 3⇥ 3 non-negative matrix
�(P+
, Q�) ⌘ 2⇡
ce+
e�L+,�
✓
p+
0
m+
c
q�0
m�c
◆�1
⇤(P+
, Q�)S(P+
, Q�),
where L+,� is the Couloumb logarithm for +� interactions. The Lorentz inner prod-
uct with signature (+���) is given by
P+
·Q� = p+
0
q�0
� p · q.
We distinguish between the standard inner product and the Lorentz inner product of
relativistic four-vectors by using capital letters P+
and Q� to denote the four-vectors.
Then, for the convenience of future analysis, we define
⇤ ⌘✓
P+
m+
c· Q�m�c
◆
2
(
✓
P+
m+
c· Q�m�c
◆
2
� 1
)�3/2
,
S ⌘(
✓
P+
m+
c· Q�m�c
◆
2
� 1
)
I3
�✓
p
m+
c� q
m�c
◆
⌦✓
p
m+
c� q
m�c
◆
+
⇢✓
P+
m+
c· Q�m�c
◆
� 1
�✓
p
m+
c⌦ q
m�c+
q
m�c⌦ p
m+
c
◆
.
This kernel is the relativtistic counterpart of the celebrated classical (non-relativistic)
Landau collision operator.
It is well known that the collision kernel � is a non-negative matrix satisfying
(2.2)3
X
i=1
�ij(P+
, Q�)
✓
qi
q�0
� pi
p+
0
◆
=3
X
j=1
�ij(P+
, Q�)
✓
qj
q�0
� pj
p+
0
◆
= 0,
10
and [47,50]
X
i,j
�ij(P+
, Q�)wiwj > 0 if w 6= d
✓
p
p+
0
� q
q�0
◆
8d 2 R.
The same is true for each other sign configuration ((+, +), (�, +), (�,�)). This prop-
erty represents the physical assumption that so-called “grazing collisions” dominate,
e.g. the change in “momentum of the colliding particles is perpendicular to their
relative velocity” [50] [p. 170]. This is also the key property used to derive the
conservation laws and the entropy dissipation below.
It formally follows from (2.2) that for number density functions g+
(p), h�(p)
Z
R3
8
>
>
>
<
>
>
>
:
0
B
B
B
@
1
p
p+
0
1
C
C
C
A
C(h+
, g�)(p) +
0
B
B
B
@
1
p
p�0
1
C
C
C
A
C(g�, h+
)(p)
9
>
>
>
=
>
>
>
;
dp = 0.
The same property holds for other sign configurations. By integrating the relativistic
Landau-Maxwell system and plugging in this identity, we deduce the local conserva-
tion of mass
Z
R3
m±
⇢
@t + cp
p±0
·rx
�
F±(t, x)dp = 0,
the local conservation of total momentum (both kinetic and electromagnetic)
Z
R3
p
✓
m+
⇢
@t + cp
p+
0
·rx
�
F+
(t, x) + m�
⇢
@t + cp
p�0
·rx
�
F�(t, x)
◆
dp
+1
4⇡@t (E(t)⇥B(t))
=1
4⇡
⇢
(B ·rx) B + (E ·rx +rx · E) E � 1
2rx
�
|B|2 + |E|2�
�
,
and the local conservation of total energy (both kinetic and electromagnetic)
Z
R3
✓
m+
p+
0
⇢
@t + cp
p+
0
·rx
�
F+
(t, x) + m�p�0
⇢
@t + cp
p�0
·rx
�
F�(t, x)
◆
dp
+1
8⇡@t
�
|E(t)|2 + |B(t)|2�
=1
4⇡{E · (rx ⇥B)�B · (rx ⇥ E)} .
11
Integration over the periodic box T3 yields the conservation of mass, total momentum
and total energy for solutions as
d
dt
Z
T3⇥R3
m+
F+
(t) =d
dt
Z
T3⇥R3
m�F�(t) = 0,
d
dt
⇢
Z
T3⇥R3
p(m+
F+
(t) + m�F�(t)) +1
4⇡
Z
T3
E(t)⇥B(t)
�
= 0,
d
dt
⇢
1
2
Z
T3⇥R3
(m+
p+
0
F+
(t) + m�p�0
F�(t)) +1
8⇡
Z
T3
|E(t)|2 + |B(t)|2�
= 0.
The entropy of the relativistic Landau-Maxwell system is defined as
H(t) ⌘Z
T3⇥R3
{F+
(t, x, p) ln F+
(t, x, p) + F�(t, x, p) ln F�(t, x, p)} dxdp � 0.
Boltzmann’s famous H-Theorem for the relativistic Landau-Maxwell system is
d
dtH(t) 0,
e.g. the entropy of solutions is non-increasing as time passes.
The global relativistic Maxwellian (a.k.a. the Juttner solution) is given by
J±(p) =exp
�
�cp±0
/(kBT )�
4⇡e±m2±ckBTK2
(m±c2/(kBT )),
where K2
(·) is the Bessel function K2
(z) ⌘ z2
3
R11
e�zt(t2 � 1)3/2dt, T is the temper-
ature and kB is Boltzmann’s constant. From the Maxwell system and the periodic
boundary condition of E(t, x), we see that ddt
R
T3 B(t, x)dx ⌘ 0. We thus have a con-
stant B such that
(2.3)1
|T3|
Z
T3
B(t, x)dx = B.
Let [·, ·] denote a column vector. We then have the following steady state solution to
the relativisitic Landau-Maxwell system
[F±(t, x, p), E(t, x), B(t, x)] = [J±, 0, B],
which minnimizes the entropy (H(t) = 0).
It is our purpose to study the e↵ects of collisions in a hot plasma and to construct
global in time classical solutions for the relativistic Landau-Maxwell system with
initial data close to the relativistic Maxwellian (Theorem 2.1). Our construction
12
implies the asymptotic stability of the relativistic Maxwellian, which is suggested by
the H-Theorem.
2.2. Main Results
We define the standard perturbation f±(t, x, p) to J± as
F± ⌘ J± +p
J±f±.
We will plug this perturbation into the Landau-Maxwell system of equations to derive
a perturbed Landau-Maxwell system for f±(t, x, p), E(t, x) and B(t, x). The two
Landau-Maxwell equations for the perturbation f = [f+
, f�] take the form⇢
@t + cp
p±0
·rx ± e±
✓
E +p
p±0
⇥B
◆
·rp
�
f± ⌥e±c
kBT
⇢
E · p
p±0
�pJ± + L±f
= ± e±c
2kBT
⇢
E · p
p±0
�
f± + �±(f, f),(2.4)
with f(0, x, p) = f0
(x, p) = [f0,+(x, p), f
0,�(x, p)]. The linear operator L±f defined
in (2.21) and the non-linear operator �±(f, f) defined in (2.23) are derived from an
expansion of the Landau collision operator (2.1). The coupled Maxwell system takes
the form
@tE � crx ⇥B = �4⇡J = �4⇡
Z
R3
⇢
e+
p
p+
0
pJ
+
f+
� e�p
p�0
pJ�f�
�
dp,
@tB + crx ⇥ E = 0,(2.5)
with constraints
rx · E = 4⇡⇢ = 4⇡
Z
R3
n
e+
pJ
+
f+
� e�p
J�f�o
dp, rx ·B = 0,(2.6)
with E(0, x) = E0
(x), B(0, x) = B0
(x). In computing the charge ⇢, we have used the
normalizationR
R3 J±(p)dp = 1
e±.
Notation: For notational simplicity, we shall use h·, ·i to denote the standard L2
inner product in R3 and (·, ·) to denote the standard L2 inner product in T3 ⇥ R3.
We define the collision frequency as the 3⇥ 3 matrix
(2.7) �ij±,⌥(p) ⌘
Z
�ij(P±, Q⌥)J⌥(q)dq.
13
These four weights (corresponding to signatures (+, +), (+,�), (�, +), (�,�)) are
used to measure the dissipation of the relativistic Landau collision term. Unless
otherwise stated g = [g+
, g�] and h = [h+
, h�] are functions which map {t � 0} ⇥
T3 ⇥ R3 ! R2. We define
hg, hi� ⌘Z
R3
��
�ij+,+ + �ij
+,��
@pj
g+
@pi
h+
+�
�ij�,� + �ij
�,+
�
@pj
g�@pi
h�
dp,
+1
4
Z
R3
�
�ij+,+ + �ij
+,�� pi
p+
0
pj
p+
0
g+
h+
dp(2.8)
+1
4
Z
R3
�
�ij�,� + �ij
�,+
� pi
p�0
pj
p�0
g�h�dp,
where in (2.8) and the rest of the paper we use the Einstein convention of implic-
itly summing over i, j 2 {1, 2, 3} (unless otherwise stated). This complicated inner
product is motivated by following splitting, which is a crucial element of the energy
method used in this paper (Lemma 2.6 and Lemma 2.8):
hLg, hi = h[L+
g, L�g], hi = hg, hi� + a “compact” term.
We will also use the corresponding L2 norms
|g|2� ⌘ hg, gi�, kgk2
� ⌘ (g, g)� ⌘Z
T3
hg, gi�dx.
We use | · |2
to denote the L2 norm in R3 and k · k to denote the L2 norm in either
T3⇥R3 or T3 (depending on whether the function depends on both (x, p) or only on
x). Let the multi-indices � and � be � = [�0, �1, �2, �3], � = [�1, �2, �3]. We use the
following notation for a high order derivative
@�� ⌘ @�0
t @�1
x1@�2
x2@�3
x3@�1
p1@�2
p2@�3
p3.
14
If each component of � is not greater than that of �’s, we denote by � �; � < �
means � �, and |�| < |�|. We also denote
0
@
�
�
1
A by C¯�� . Let
|||f |||2(t) ⌘X
|�|+|�|N
||@��f(t)||2,
|||f |||2�(t) ⌘X
|�|+|�|N
||@��f(t)||2�,
|||[E, B]|||2(t) ⌘X
|�|N
||[@�E(t), @�B(t)]||2.
It is important to note that our norms include the temporal derivatives. For a function
independent of t, we use the same notation but we drop the (t). The above norms
and their associated spaces are used throughout the paper for arbitrary functions.
We further define the high order energy norm for a solution f(t, x, p), E(t, x) and
B(t, x) to the relativistic Landau-Maxwell system (2.4) and (2.5) as
(2.9) E(t) ⌘ 1
2|||f |||2(t) + |||[E, B]|||2(t) +
Z t
0
|||f |||2�(s)ds.
Given initial datum [f0
(x, p), E0
(x), B0
(x)], we define
E(0) =1
2|||f
0
|||2 + |||[E0
, B0
]|||2,
where the temporal derivatives of [f0
, E0
, B0
] are defined naturally through equations
(2.4) and (2.5). The high order energy norm is consistent at t = 0 for a smooth
solution and E(t) is continuous (Theorem 2.6).
Assume that initially [F0
, E0,B0
] has the same mass, total momentum and total
energy as the steady state [J±, 0, B], then we can rewrite the conservation laws in
terms of the perturbation [f, E, B]:Z
T3⇥R3
m+
f+
(t)p
J+
⌘Z
T3⇥R3
m�f�(t)p
J� ⌘ 0,(2.10)
Z
T3⇥R3
pn
m+
f+
(t)p
J+
+ m�f�(t)p
J�o
⌘ � 1
4⇡
Z
T3
E(t)⇥B(t),(2.11)
Z
T3⇥R3
m+
p+
0
f+
(t)p
J+
+ m�p�0
f�(t)p
J� ⌘ �1
8⇡
Z
T3
|E(t)|2 + |B(t)� B|2.(2.12)
We have used (2.3) for the normalized energy conservation (2.12).
15
The e↵ect of this restriction is to guarantee that a solution can only converge to the
specific relativistic Maxwellian that we perturb away from (if the solution converges
to a relativistic Maxwellian). The value of the steady state B is also defined by the
initial conditions (2.3).
We are now ready to state our main results:
Theorem 2.1. Fix N , the total number of derivatives in (2.9), with N � 4.
Assume that [f0
, E0
, B0
] satisfies the conservation laws (2.10), (2.11), (2.12) and the
constraint (2.6) initially. Let
F0,±(x, p) = J± +
pJ±f
0,±(x, p) � 0.
There exist C0
> 0 and M > 0 such that if
E(0) M,
then there exists a unique global solution [f(t, x, p), E(t, x), B(t, x)] to the perturbed
Landau-Maxwell system (2.4), (2.5) with (2.6). Moreover,
F±(t, x, p) = J± +p
J±f±(t, x, p) � 0
solves the relativistic Landau-Maxwell system and
sup0s1
E(s) C0
E(0).
Remarks:
• These solutions are C1, and in fact Ck, for N large enough.
• SinceR1
0
|||f |||2�(t)dt < +1, f(t, x, p) gains one momentum derivative over
it’s initial data and |||f |||2�(t) ! 0 in a certain sense.
• Further, Lemma 2.5 together with Lemma 2.13 imply that
X
|�|N�1
�
||@�E(t)||+ ||@�{B(t)� B}||
CX
|�|N
||@�f(t)||�.
Therefore, except for the highest order derivatives, the field also converges.
• It is an interesting open question to determine the asymptotic behavior of
the highest order derivatives of the electromagnetic field.
16
Recently, global in time solutions to the related classical Vlasov-Maxwell-Boltzmann
equation were constructed by the second author in [39]. The Boltzmann equation is a
widely accepted model for binary interactions in a dilute gas, however it fails to hold
for a dilute plasma in which grazing collisions dominate.
The following classical Landau collision operator (with normalized constants) was
designed to model such a plasma:
Ccl(F�, F+
) ⌘ rv ·⇢
Z
R3�(v � v0) {rvF�(v)F
+
(v0)� F�(v)rv0F+
(v0)} dv0�
.
The non-negative 3⇥ 3 matrix is
(2.13) �ij(v) =
⇢
�ij �vivj
|v|2�
1
|v| .
Unfortunately, because of the crucial hard sphere assumption, the construction in [39]
fails to apply to a non-relativistic Coulombic plasma interacting with it’s electromag-
netic field. The key problem is that the classical Landau collision operator, which
was studied in detail in [38], o↵ers weak dissipation of the formR
R3(1 + |v|)�1|f |2dv.
The global existence argument in Section 2.6 (from [39]) does not work because of
this weak dissipation. Further, the unbounded velocity v, which is inconsistent with
Einstein’s theory of special relativity, in particular makes it impossible to control a
nonlinear term like {E · v} f± in the classical theory.
On the other hand, in the relativistic case our key observation is that the cor-
responding nonlinear term c�
E · p/p±0
f± can be easily controlled by the dissipa-
tion because |cp/p±0
| c and the dissipation in the relativisitc Landau operator isR
R3 |f |2dp (Lemma 2.5 and Lemou [47]).
However, it is well-known that the relativity e↵ect can produce severe mathe-
matical di�culties. Even for the related pure relativistic Boltzmann equation, global
smooth solutions were only constructed in [32,33].
The first new di�culty is due to the complexity of the relativistic Landau collision
kernel �(P+
, Q�). Since
P+
m+
c· Q�m�c
� 1 ⇠ 1
2c
�
�
�
�
p
m+
� q
m�
�
�
�
�
2
whenP
+
m+
c⇡ Q�
m�c,
17
the kernel in (2.1) has a first order singularity. Hence it can not absorb many deriva-
tives in high order estimates (Lemma 2.7 and Theorem 2.4). The same issue exists
for the classical Landau kernel �(v � v0), but the obvious symmetry makes it easy to
express v derivatives of � in terms of v0 derivatives. It is then possible to integrate
by parts and move derivatives o↵ the singular kernel in the estimates of high order
derivatives. On the contrary, no apparent symmetry exists beween p and q in the
relativistic case. We overcome this severe di�culty with the splitting
@pj
�ij(P+
, Q�) = �q�0
p+
0
@qj
�ij(P+
, Q�) +
✓
@pj
+q�0
p+
0
@qj
◆
�ij(P+
, Q�),
where the operator⇣
@pj
+ q�0p+0@q
j
⌘
does not increase the order of the singularity mainly
because✓
@pj
+q�0
p+
0
@qj
◆
P+
·Q� = 0.
This splitting is crucial for performing the integration by parts in all of our estimates
(Lemma 2.2 and Theorem 2.3). We believe that such an integration by parts technique
should shed new light on the study of the relativistic Boltzmann equation.
As in [38,39], another key point in our construction is to show that the linearized
collision operator L is in fact coercive for solutions of small amplitude to the full
nonlinear system (2.4), (2.5) and (2.6):
Theorem 2.2. Let [f(t, x, p), E(t, x), B(t, x)] be a classical solution to (2.4) and
(2.5) satisfying (2.6), (2.10), (2.11) and (2.12). There exists M0
, �0
= �0
(M0
) > 0
such that if
(2.14)X
|�|N
⇢
1
2||@�f(t)||2 + ||@�E(t)||2 + ||@�B(t)||2
�
M0
,
thenX
|�|N
(L@�f(t), @�f(t)) � �0
X
|�|N
||@�f(t)||2�.
Theorem 2.2 is proven through a careful study of the macroscopic equations (2.98)
- (2.102). These macroscopic equations come from a careful study of solutions f to
the perturbed relativistic Landau-Maxwell system (2.4), (2.5) with (2.6) projected
18
onto the null space N of the linearized collision operator L = [L+
, L�] defined in
(2.21).
As expected from the H-theorem, L is non-negative and for every fixed (t, x) the
null space of L is given by the six dimensional space (1 i 3)
(2.15) N ⌘ span{[p
J+
, 0], [0,p
J�], [pi
pJ
+
, pi
pJ�], [p+
0
pJ
+
, p�0
pJ�]}.
This is shown in Lemma 2.1. We define the orthogonal projection from L2(R3
p) onto
the null space N by P. We then decompose f(t, x, p) as
f = Pf + {I�P}f.
We call Pf = [P+
f,P�f ] 2 R2 the hydrodynamic part of f and {I�P}f =
[{I�P}+
f, {I�P}�f ] is called the microscopic part. By separating its linear and
nonlinear part, and using L±{Pf} = 0, we can express the hydrodynamic part of f
through the microscopic part up to a higher order term h(f):
(2.16)
⇢
@t + cp
p±0
·rx
�
P±f ⌥ e±c
kBT
⇢
E · p
p±0
�pJ± = l±({I�P}f) + h±(f),
where
l±({I�P}f) ⌘ �⇢
@t +p
p±0
·rx
�
{I�P}±f + L± {{I�P}f} ,(2.17)
h±(f) ⌘ ⌥e±
✓
E +p
p±0
⇥B
◆
·rpf±
± e±c
2kBT
⇢
E · p
p±0
�
f± + �±(f, f).(2.18)
We further expand P±f as a linear combination of the basis in (2.15)
(2.19) P±f ⌘(
a±(t, x) +3
X
j=1
bj(t, x)pj + c(t, x)p±0
)
pJ±.
A precise definition of these coe�cients will be given in (2.94). The relativistic system
of macroscopic equations (2.98) - (2.102) are obtained by plugging (2.19) into (2.16).
These macroscopic equations for the coe�cients in (2.19) enable us to show that
there exists a constant C > 0 such that solutions to (2.4) which satisfy the smallness
19
constraint (2.14) (for M0
> 0 small enough) will also satisfy
(2.20)X
|�|N
{||@�a±||+ ||@�b||+ ||@�c||} C(M0
)X
|�|N
||{I�P}@�f(t)||�.
This implies Theorem 2.2 since kPfk� is trivially bounded above by the l.h.s. (Propo-
sition 2.2) and L is coercive with respect to {I�P}@�f(t) (Lemma 2.8).
Since our smallness assumption (2.14) involves no momentum derivatives, in prov-
ing (2.20) the presence of momentum derivatives in the collision operator (2.1) causes
another serious mathematical di�culty. We develop a new estimate (Theorem 2.5)
which involves purely spatial derivatives of the linear term (2.21) and the nonlinear
term (2.23) to overcome this di�culty.
To the best of the authors’ knowledge, until now there were no known solutions
for the relativistic Landau-Maxwell system. However in 2000, Lemou [47] studied the
linearized relativistic Landau equation with no electromagnetic field. We will use one
of his findings (Lemma 2.5) in the present work.
For the classical Landau equation, the 1990’s have seen the first solutions. In 1994,
Zhan [72] proved local existence and uniqueness of classical solutions to the Landau-
Poisson equation (B ⌘ 0) with Coulomb potential and a smallness assumption on
the initial data at infinity. In the same year, Zhan [73] proved local existence of weak
solutions to the Landau-Maxwell equation with Coulomb potential and large initial
data.
On the other hand, in the absence of an electromagnetic field we have the following
results. In 2000, Desvillettes and Villani [17] proved global existence and uniqueness of
classical solutions for the spatially homogeneous Landau equation for hard potentials
and a large class of initial data. In 2002, the second author [38] constructed global
in time classical solutions near Maxwellian for a general Landau equation (both hard
and soft potentials) in a periodic box based on a nonlinear energy method.
Our paper is organized as follows. In section 2.3 we establish linear and nonlinear
estimates for the relativistic Landau collision operator. In section 2.4 we construct
20
local in time solutions to the relativistic Landau-Maxwell system. In section 2.5 we
prove Theorem 2.2. And in section 2.6 we extend the solutions to T = 1.
Remark 2.1. It turns out that the presence of the physical constants do not cause
essential mathematical di�culties. Therefore, for notational simplicity, after the proof
of Lemma 2.1 we will normalize all constants in the relativistic Landau-Maxwell sys-
tem (2.4), (2.5) with (2.6) and in all related quantities to be one.
2.3. The Relativistic Landau Operator
Our main results in this section include the crucial Theorem 2.3, which allows us
to express p derivatives of �(P, Q) in terms of q derivatives of �(P, Q). This is vital for
establishing the estimates found at the end of the section (Lemma 2.7, Theorem 2.4
and Theorem 2.5). Other important results include the equivalence of the norm | · |�with the standard Sobolev space norm for H1 (Lemma 2.5) and a weak formulation
of compactness for K which is enough to prove coercivity for L away from the null
space N (Lemma 2.8). We also compute the sum of second order derivatives of the
Landau kernel (Lemma 2.3).
We first introduce some notation. Using (2.2), we observe that quadratic collision
operator (2.1) satisfies
C(J+
, J+
) = C(J+
, J�) = C(J�, J+
) = C(J�, J�) = 0.
Therefore, the linearized collision operator Lg is defined by
(2.21) Lg = [L+
g, L�g], L±g ⌘ �A±g �K±g,
where
A+
g ⌘ J�1/2
+
C(p
J+
g+
, J+
) + J�1/2
+
C(p
J+
g+
, J�),
A�g ⌘ J�1/2
� C(p
J�g�, J�) + J�1/2
� C(p
J�g�, J+
),
K+
g ⌘ J�1/2
+
C(J+
,p
J+
g+
) + J�1/2
+
C(J+
,p
J�g�),(2.22)
K�g ⌘ J�1/2
� C(J�,p
J�g�) + J�1/2
� C(J�,p
J+
g+
).
21
And the nonlinear part of the collision operator (2.1) is defined by
�(g, h) = [�+
(g, h),��(g, h)],
where
�+
(g, h) ⌘ J�1/2
+
C(p
J+
g+
,p
J+
h+
) + J�1/2
+
C(p
J+
g+
,p
J�h�),(2.23)
��(g, h) ⌘ J�1/2
� C(p
J�g�,p
J�h�) + J�1/2
� C(p
J�g�,p
J+
h+
).
We will next derive the null space (2.15) of the linear operator in the presence of all
the physical constants.
Lemma 2.1. hLg, hi = hLh, gi, hLg, gi � 0. And Lg = 0 if and only if g = Pg.
Proof. From (2.21) we split hLg, hi, with Lg = [L+
g, L�g], as
�Z
R3
h+pJ
+
{C(p
J+
g+
, J+
) + C(J+
,p
J+
g+
)}dp
�Z
R3
⇢
h+pJ
+
{C(p
J+
g+
, J�) + C(J+
,p
J�g�)}�
dp(2.24)
�Z
R3
⇢
h�pJ�
{C(p
J�g�, J+
) + C(J�,p
J+
g+
)}�
dp
�Z
R3
h�pJ�{C(p
J�g�, J�) + C(J�,p
J�g�)}dp.
We use the fact that @qi
J�(q) = � ck
B
Tqi
q�0J�(q) and @p
i
J1/2
+
(p) = � ck
B
Tp
i
p+0J
+
(p) as well
as the null space of � in (2.2) to show that
C(J1/2
+
g+
, J�)
= @pi
Z
R3
�ij(P+
, Q�)J�(q)J1/2
+
(p)
⇢✓
qi
q�0
� pi
2p+
0
◆
g+
(p) + @pj
g+
(p)
�
dq
= @pi
Z
R3
�ij(P+
, Q�)J�(q)J1/2
+
(p)
⇢
pi
2p+
0
g+
(p) + @pj
g+
(p)
�
dq
= @pi
Z
R3
�ij(P+
, Q�)J�(q)J+
(p)@pj
(J�1/2
+
g+
(p))dq
22
And similarly
C(J�, J1/2
+
g+
)
= �@pi
Z
R3
�ij(P�, Q+
)J�(p)J1/2
+
(q)
⇢✓
pi
p�0
� qi
2q+
0
◆
g+
(q) + @qj
g+
(q)
�
dq
= �@pi
Z
R3
�ij(P�, Q+
)J�(p)J1/2
+
(q)
⇢
qi
2q+
0
g+
(q) + @qj
g+
(q)
�
dq
= �@pi
Z
R3
�ij(P�, Q+
)J�(p)J+
(q)@qj
(J�1/2
+
g+
(q))dq.(2.25)
Similar expressions hold by exchanging the + terms and the � terms in the appropri-
ate places. For the first term in (2.24), we integrate by parts over p variables on the
first line, then relabel the variables switching p and q on the second line and finally
adding them up on the last line to obtain
=
ZZ
�ij(P+
, Q+
)J+
(p)J+
(q)@pi
(h+
J�1/2
+
(p))
⇥{@pj
(g+
J�1/2
+
(p))� @qj
(g+
J�1/2
+
(q))}dpdq
=
ZZ
�ij(P+
, Q+
)J+
(p)J+
(q)@qi
(h+
J�1/2
+
(q))
⇥{@qj
(g+
J�1/2
+
(q))� @pj
(g+
J�1/2
+
(p))}dpdq
=1
2
ZZ
�ij(P+
, Q+
)J+
(p)J+
(q){@pi
(h+
J�1/2
+
(p))� @qi
(h+
J�1/2
+
(q))}
⇥{@pj
(g+
J�1/2
+
(p))� @qj
(g+
J�1/2
+
(q))}dpdq.
By (2.2) the first term in (2.24) is symmetric and � 0 if h = g. The fourth term can
be treated similarly (with + replaced by � everywhere. We combine the second and
third terms in (2.24); again we integrate by parts over p variables to compute
=
ZZ
�ij(P+
, Q�)J+
(p)J�(q)@pi
(h+
J�1/2
+
(p))
⇥{@pj
(g+
J�1/2
+
(p))� @qj
(g�J�1/2
� (q))}dpdq
+
ZZ
�ij(P�, Q+
)J�(p)J+
(q)@pi
(h�J�1/2
� (p))
⇥{@pj
(g�J�1/2
� (p))� @qj
(g+
J�1/2
+
(q))}dpdq.
23
We switch the role of p and q in the second term to obtain
=
ZZ
�ij(P+
, Q�)J+
(p)J�(q){@pi
(h+
J�1/2
+
(p))� @qi
(h�J�1/2
� (q))}
⇥{@pj
(g+
J�1/2
+
(p))� @qj
(g�J�1/2
� (q))}dpdq.
Again by (2.2) this piece of the operator is symmetric and � 0 if g = h. We therefore
conclude that L is a non-negative symmetric operator.
We will now determine the null space (2.15) of the linear operator. Assume Lg = 0.
From hLg, gi = 0 we deduce, by (2.2), that there are scalar functions ⇣l(p, q) (l = ±)
such that
@pi
(glJ�1/2
l (p))� @qi
(glJ�1/2
l (q)) ⌘ ⇣l(p, q)
✓
pi
pl0
� qi
ql0
◆
, i 2 {1, 2, 3}.
Setting q = 0, @pi
(glJ�1/2
l (p)) = ⇣l(p, 0) pi
pl
0+ bli. By replacing p by q and subtracting
we obtain
@pi
(glJ�1/2
l (p))� @qi
(glJ�1/2
l (q)) = ⇣l(p, 0)pi
pl0
� ⇣l(q, 0)qi
ql0
= ⇣l(p, 0)
✓
pi
pl0
� qi
ql0
◆
+ (⇣l(p, 0)� ⇣l(q, 0))qi
ql0
.
We deduce, again by (2.2), that ⇣l(p, 0)� ⇣l(q, 0) = 0 and therefore that ⇣l(p, 0) ⌘ cl
(a constant). We integrate @pi
(glJ�1/2
l (p)) = clp
i
pl
0+ bli to obtain
gl = {agl +
3
X
i=1
bglipj + cg
l pl0
}J1/2
l .
Here agl , bg
lj and cgl are constants with respect to p (but could be functions of t and
x). Moreover, we deduce from the middle terms in (2.24) as well as (2.2) that
@pi
(g+
J�1/2
+
(p))� @qi
(g�J�1/2
� (q)) ⌘ ⇣(p, q)
✓
pi
p+
0
� qi
q�0
◆
.
Therefore bg+i � bg
�i + cg+
pi
p+0� cg
�qi
q�0= ⇣(p, q)
⇣
pi
p0� q
i
q0
⌘
. We conclude
bg+i ⌘ bg
�i, i = 1, 2, 3;
cg+
⌘ cg�.
24
That means g(t, x, p) 2 N as in (2.15), so that g = Pg. Conversely, L{Pg} = 0 by a
direct calculation. ⇤
For notational simplicity, as in Remark 2.1, we will normalize all the constants to
be one. Accordingly, we write p0
=p
1 + |p|2, P = (p0
, p), and the collision kernel
�(P, Q) takes the form
�(P, Q) ⌘ ⇤(P, Q)
p0
q0
S(P, Q),(2.26)
where
⇤ ⌘ (P ·Q)2
�
(P ·Q)2 � 1 �3/2
,
S ⌘�
(P ·Q)2 � 1
I3
� (p� q)⌦ (p� q) + {(P ·Q)� 1} (p⌦ q + q ⌦ p) .
We normalize the relativistic Maxwellian as
J(p) ⌘ J+
(p) = J�(p) = e�p0 .
We further normalize the collision freqency
(2.27) �ij±,⌥(p) = �ij(p) =
Z
�ij(P, Q)J(q)dq,
and the inner product h·, ·i� takes the form
hg, hi� ⌘ 2
Z
R3
�ij�
@pj
g+
@pi
h+
+ @pj
g�@pi
h�
dp,
+1
2
Z
R3
�ij pi
p0
pj
p0
{g+
h+
dp + g�h�} dp.(2.28)
The norms are, as before, naturally built from this normalized inner product.
The normalized vector-valued Landau-Maxwell equation for the perturbation f in
(2.4) now takes the form
⇢
@t +p
p0
·rx + ⇠
✓
E +p
p0
⇥B
◆
·rp
�
f �⇢
E · p
p0
�pJ⇠
1
+ Lf
=⇠
2
⇢
E · p
p0
�
f + �(f, f).(2.29)
25
with f(0, x, v) = f0
(x, v), ⇠1
= [1,�1], and the 2⇥ 2 matrix ⇠ is diag(1,�1). Further,
the normalized Maxwell system in (2.5) and (2.6) takes the form
@tE �rx ⇥B = �J = �Z
R3
p
p0
pJ(f
+
� f�)dp, @tB +rx ⇥ E = 0,(2.30)
rx · E = ⇢ =
Z
R3
pJ(f
+
� f�)dp, rx ·B = 0,(2.31)
with E(0, x) = E0
(x), B(0, x) = B0
(x).
We have a basic (but useful) inequality taken from Glassey & Strauss [32].
Proposition 2.1. Let p, q 2 R3 with P = (p0
, p) and Q = (q0
, q) then
(2.32)|p� q|2 + |p⇥ q|2
2p0
q0
P ·Q� 1 1
2|p� q|2.
This will inequality will be used many times for estimating high order derivatives of
the the collision kernel.
Notice that
✓
@pi
+q0
p0
@qi
◆
P ·Q =
✓
@pi
+q0
p0
@qi
◆
(p0
q0
� p · q)
=pi
p0
q0
� qi +q0
p0
✓
qi
q0
p0
� pi
◆
= 0.
This is the key observation which allows us do analysis on the relativistic Landau
Operator (Lemma 2.2). We define the following relativistic di↵erential operator
(2.33) ⇥↵(p, q) ⌘✓
@p3 +q0
p0
@q3
◆↵3✓
@p2 +q0
p0
@q2
◆↵2✓
@p1 +q0
p0
@q1
◆↵1
.
Unless otherwise stated, we omit the p, q dependence and write ⇥↵ = ⇥↵(p, q). Note
that the three terms in ⇥↵ do not commute (and we choose this order for no special
reason).
We will use the following splitting many times in the rest of this section,
(2.34) A = {|p� q|+ |p⇥ q| � [|p|+ 1]/2}, B = {|p� q|+ |p⇥ q| [|p|+ 1]/2}.
The set A is designed to be away from the first order singularity in collision kernel
�(P, Q) (Proposition 2.1). And the set B contains a �(P, Q) singularity ((2.26) and
26
(2.32)) but we will exploit the fact that we can compare the size of p and q. We now
develop crucial estimates for ⇥↵�(P, Q):
Lemma 2.2. For any multi-index ↵, the Lorentz inner product of P and Q is in
the null space of ⇥↵,
⇥↵(P ·Q) = 0.
Further, recalling (2.26), for p and q on the set A we have the estimate
(2.35) |⇥↵(p, q)�(P, Q)| Cp�|↵|0
q6
0
.
And on B,
(2.36)1
6q0
p0
6q0
.
Using this inequality, we have the following estimate on B
(2.37) |⇥↵(p, q)�(P, Q)| Cq7
0
p�|↵|0
|p� q|�1.
Proof. Let ei (i = 1, 2, 3) be an element of the standard basis in R3. We have
seen that ⇥ei
(P ·Q) = 0. And the general case follows from a simple induction over
|↵|.
By (2.26) and (2.33), we can now write
⇥↵(p, q)�ij(P, Q) = ⇤(P, Q)⇥↵(p, q)
✓
Sij(p, q)
p0
q0
◆
,
where
⇥↵
✓
Sij(p, q)
p0
q0
◆
=�
(P ·Q)2 � 1
⇥↵ {�ij/(p0
q0
)}
+(P ·Q� 1)⇥↵ {(piqj + pjqi)/(p0
q0
)}(2.38)
�⇥↵ {(pi � qi)(pj � qj)/(p0
q0
)} .
We will break up this expression and estimate the di↵erent pieces.
Using (2.33), the following estimates are straight forward
|⇥↵ {�ij/(p0
q0
)}| Cq�1
0
p�1�|↵|0
,(2.39)
|⇥↵ {(piqj + pjqi)/(p0
q0
)}| Cp�|↵|0
.(2.40)
27
On the other hand, we claim that
(2.41) |⇥↵ {(pi � qi)(pj � qj)/(p0
q0
)}| C|p� q|2
p0
q0
p�|↵|0
.
This last estimate is not so trivial because only a lower order estimate of |p � q| is
expected after applying even a first order derivative like ⇥ei
. The key observation is
that✓
@pi
+q0
p0
@qi
◆
(pi � qi)(pj � qj) =
✓
1� q0
p0
◆
(pj � qj),
and the r.h.s. is again second order. Therefore the operator ⇥↵ can maintain the
order of the cancellation.
Proof of claim: To prove (2.41), it is su�cient to show that for any multi-index ↵
and any i, j, k, l 2 {1, 2, 3} there exists a smooth function G↵,ijkl (p, q) satisfying
(2.42) ⇥↵ {(pi � qi)(pj � qj)/(p0
q0
)} =3
X
k,l=1
(pk � qk)(pl � ql)G↵,ijkl (p, q)
as well as the decay
(2.43)�
�@⌫1@q⌫2
G↵,ijkl (p, q)
�
� Cq�1�|⌫2|0
p�1�|↵|�|⌫1|0
,
which holds for any multi-indices ⌫1
, ⌫2
. We prove (2.42) with (2.43) by a simple
induction over |↵|.
If |↵| = 0, we define
G0,ijkl (p, q) =
�ki�ljq0
p0
.
The decay (2.43) for G0,ijkl (p, q) is straight forward to check. And (2.42) holds trivially
for |↵| = 0.
Assume the (2.42) with (2.43) holds for |↵| n. To conclude the proof, let
|↵0| = n + 1 and write ⇥↵0 = ⇥em
⇥↵ for some multi-index ↵ with
m = max{j : (↵0)j > 0}.28
This specification of m is needed because of our chosen ordering of the three di↵er-
ential operators in (2.33), which don’t commute. Recalling (2.33),
⇥em
(pk � qk) = �km
✓
1� q0
p0
◆
.
From the induction assumption and the last display, we have
⇥↵0 {(pi � qi)(pj � qj)/(p0
q0
)}
= ⇥em
3
X
k,l=1
(pk � qk)(pl � ql)G↵,ijkl (p, q),
=
✓
1� q0
p0
◆
3
X
k=1
(pk � qk)�
G↵,ijmk (p, q) + G↵,ij
km (p, q)
+3
X
k,l=1
(pk � qk)(pl � ql)⇥em
G↵,ijkl (p, q).
We compute
1� q0
p0
=p
0
� q0
p0
=p2
0
� q2
0
p0
(p0
+ q0
)=
(p� q) · (p + q)
p0
(p0
+ q0
)=
P
l(pl � ql)(pl + ql)
p0
(p0
+ q0
).
We plug this display into the one above it to obtain (2.42) for ↵0 with the new
coe�cients
G↵0,ijkl (p, q) ⌘ ⇥e
m
G↵,ijkl (p, q) +
�
G↵,ijmk (p, q) + G↵,ij
km (p, q)
(pl + ql)
p0
(p0
+ q0
).
We check that G↵0,ijkl (p, q) satisfies (2.43) using the Leibnitz di↵erentiation formula as
well as the induction assumption (2.43). This establishes the claim (2.41).
With the estimates (2.39), (2.40) and (2.41) in hand, we return to establishing
(2.35) and (2.37). We plug the estimates (2.39), (2.40) and (2.41) into ⇥↵�ij(P, Q)
from (2.38) to obtain that
�
�⇥↵�ij(P, Q)
�
� Cp�|↵|0
(P ·Q)2
�
(P ·Q)2 � 1 �3/2
(P ·Q)2 � 1
p0
q0
+Cp�|↵|0
(P ·Q)2
�
(P ·Q)2 � 1 �3/2
(P ·Q� 1)(2.44)
+Cp�|↵|0
(P ·Q)2
�
(P ·Q)2 � 1 �3/2
|p� q|2p
0
q0
.
We will use this estimate twice to get (2.35) and (2.37).
29
We first establish (2.35). On the set A we have
2|p� q|2 + 2|p⇥ q|2 � (|p� q|+ |p⇥ q|)2 � 1
4p2
0
+|p|2� 1
4p2
0
.
From (2.32) and the last display we have
P ·Q + 1 � P ·Q� 1 � 1
16
p0
q0
.
From the Cauchy-Schwartz inequality we also have
0 P ·Q� 1 P ·Q p0
q0
+ |p · q| 2p0
q0
.
We plug these last two inequalities (one at a time) into (2.44) to obtain
�
�⇥↵�ij(p, q)
�
� C(P ·Q)2
�
(P ·Q)2 � 1 �3/2
p�|↵|0
⇢
p2
0
q2
0
+ p2
0
q2
0
+ p2
0
q2
0
p0
q0
�
C(p0
q0
)2
�
(P ·Q)2 � 1 �3/2
p�|↵|0
p0
q0
C(p0
q0
)3 {P ·Q� 1}�3 p�|↵|0
C(p0
q0
)3
✓
p0
q0
◆�3
p�|↵|0
.
We move on to establishing (2.36). If |p| 1, then p0
2 2q0
. Assume |p| � 1,
using B we compute
q0
� |q| � |p|� |p� q| � 1
2|p|� 1
2� 1
4p
0
� 1
2.
Therefore, p0
6q0
on B. For the other half of (2.36),
q0
p0
+ |p� q| 3
2p
0
+1
2 2p
0
.
We move on to establishing (2.37). On the set B we have a first order singularity.
Also (2.32) tells us
|p� q|22p
0
q0
P ·Q� 1 1
2|p� q|2.
30
We plug this into (2.44) to observe that on B we have
�
�⇥↵�ij(p, q)
�
� C(P ·Q)2
�
(P ·Q)2 � 1 �3/2
p�|↵|0
⇥⇢
(P ·Q + 1)|p� q|2 + |p� q|2 + |p� q|2q0
p0
q0
p0
�
C(P ·Q)2
�
(P ·Q)2 � 1 �3/2
p�|↵|0
|p� q|2
C(p0
q0
)2
�
(P ·Q)2 � 1 �3/2
p�|↵|0
|p� q|2
C(p0
q0
)2(p0
q0
)3/2|p� q|�1 {P ·Q + 1}�3/2 p�|↵|0
C(p0
q0
)7/2|p� q|�1p�|↵|0
.
We achieve the last inequality because (2.32) says P ·Q � 1. ⇤
Next, let µ(p, q) be an arbitrary smooth scalar function which decay’s rapidly at
infinity. We consider the following integralZ
R3
�ij(P, Q)J1/2(q)µ(p, q)dq.
Both the linear term L and the nonlinear term � are of this form (Lemma 2.6). We
develop a new integration by parts technique.
Theorem 2.3. Given |�| > 0, we have
@�
Z
R3
�ij(P, Q)J1/2(q)µ(p, q)dq
=X
�1+�2+�3�
Z
R3
⇥�1�ij(P, Q)J1/2(q)@q
�2@�3µ(p, q)'�
�1,�2,�3(p, q)dq(2.45)
where '��1,�2,�3
(p, q) is a smooth function which satisfies
(2.46)�
�
�
@q⌫1@⌫2'
��1,�2,�3
(p, q)�
�
�
Cq|�|�|⌫1|0
p|�1|+|�3|�|�|�|⌫2|0
,
for all multi-indices ⌫1
and ⌫2
.
Proof. We prove (2.45) by an induction over the number of derivatives |�|.
Assume � = ei (i = 1, 2, 3). We write
(2.47) @pi
= �q0
p0
@qi
+
✓
@pi
+q0
p0
@qi
◆
= �q0
p0
@qi
+ ⇥ei
31
Instead of hitting �ij(P, Q) with @pi
, we apply the r.h.s. term above and integrate by
parts over � q0
p0@q
i
to obtain
@pi
Z
R3
�ij(P, Q)J1/2(q)µ(p, q)dq
=
Z
R3
�ij(P, Q)J1/2(q)@pi
µ(p, q)dq
+
Z
R3
�ij(P, Q)J1/2(q)q0
p0
@qi
µ(p, q)dq
+
Z
R3
�ij(P, Q)J1/2(q)
✓
qi
q0
p0
� qi
2p0
◆
µ(p, q)dq
+
Z
R3
⇥ei
�ij(P, Q)J1/2(q)µ(p, q)dq.
We can write the above in the form (2.45) with the coe�cients given by
(2.48) �ei
0,0,0(p, q) =qi
q0
p0
� qi
2p0
, �ei
ei
,0,0(p, q) = 1, �ei
0,ei
,0(p, q) =q0
p0
, �ei
0,0,ei
(p, q) = 1.
And define the rest of the coe�cients to be zero. Note that these coe�cients satisfy
the decay (2.46). This establishes the first step in the induction.
Assume the result holds for all |�| n. Fix an arbitrary �0 such that |�0| = n + 1
and write @�0 = @pm
@� for some multi-index � and
m = max{j : (�0)j > 0}.
This specification of m is needed because of our chosen ordering of the three di↵er-
ential operators in (2.33), which don’t commute.
By the induction assumption
@�0
Z
R3
�ij(P, Q)J1/2(q)µ(p, q)dq
=X
¯�1+
¯�2+
¯�3�
@pm
Z
R3
⇥¯�1�ij(P, Q)J1/2(q)@q
¯�2@
¯�3µ(p, q)'�
¯�1,¯�2,¯�3(p, q)dq
32
We approach applying the last derivative the same as the |�| = 1 case above. We
obtain
=X
Z
R3
⇥¯�1�ij(P, Q)J1/2(q)'�
¯�1,¯�2,¯�3(p, q)@p
m
@q¯�2@
¯�3µ(p, q)dq(2.49)
+X
Z
R3
⇥¯�1�ij(P, Q)J1/2(q)'�
¯�1,¯�2,¯�3(p, q)
q0
p0
@qm
@q¯�2@
¯�3µ(p, q)dq(2.50)
+X
Z
R3
⇥em
⇥¯�1�ij(P, Q)J1/2(q)@q
¯�2@
¯�3µ(p, q)'�
¯�1,¯�2,¯�3(p, q)dq(2.51)
+X
Z
R3
⇥¯�1�ij(P, Q)J1/2(q)@q
¯�2@
¯�3µ(p, q)
⇥✓
@pm
+q0
p0
@qm
+qm
p0
q0
� qm
2p0
◆
'�¯�1,¯�2,¯�3
(p, q)dq,(2.52)
where the unspecified summations above are over �1
+ �2
+ �3
�. We collect all
the terms above with the same order of di↵erentiation to obtain
=X
�1+�2+�3�0
Z
R3
⇥�1�ij(P, Q)J1/2(q)@q
�2@�3µ(p, q)'�0
�1,�2,�3(p, q)dq,
where the functions '�0
�1,�2,�3(p, q) are defined naturally as the coe�cient in front of
each term of the form ⇥�1�ij(P, Q)J1/2(q)@q
�2@�3µ(p, q) and we recall that �0 = �+em.
We check (2.46) by comparing the decay with the order of di↵erentiation in each
of the four terms (2.49-2.52). For (2.49), the order of di↵erentiation is
�1
= �1
, �2
= �2
, �3
= �3
+ em.
And by the induction assumption,
�
�
�
@q⌫1@⌫2'
�¯�1,¯�2,¯�3
(p, q)�
�
�
Cq|�|�|⌫1|0
p|¯�1|+|¯�3|�|�|�|⌫2|
0
,
Cq|�+em
|�|⌫1|0
p|�1|+|¯�3+em
|�|�+ei
|�|⌫2|0
= Cq|�0|�|⌫1|
0
p|�1|+|�3|�|�0|�|⌫2|0
.
This establishes (2.46) for (2.49).
For (2.50), the order of di↵erentiation is
�1
= �1
, �2
= �2
+ em, �3
= �3
.
33
And by the induction assumption as well as the Leibnitz rule,�
�
�
�
@q⌫1@⌫2
✓
q0
p0
'�¯�1,¯�2,¯�3
(p, q)
◆
�
�
�
�
Cq|�|+1�|⌫1|0
p|¯�1|+|¯�3|�|�|�1�|⌫2|
0
,
= Cq|�0|�|⌫1|
0
p|�1|+|�3|�|�0|�|⌫2|0
.
This establishes (2.46) for (2.50).
For (2.51), the order of di↵erentiation is
�1
= �1
+ em, �2
= �2
, �3
= �3
.
And by the induction assumption,
�
�
�
@q⌫1@⌫2'
�¯�1,¯�2,¯�3
(p, q)�
�
�
Cq|�|�|⌫1|0
p|¯�1|+|¯�3|�|�|�|⌫2|
0
,
Cq|�0|�|⌫1|
0
p|�1|+|�3|�|�0|�|⌫2|0
.
This establishes (2.46) for (2.51).
For (2.52), the order of di↵erentiation is
�1
= �1
, �2
= �2
, �3
= �3
.
And by the induction assumption as well as the Leibnitz rule,�
�
�
�
@q⌫1@⌫2
⇢✓
@pm
+q0
p0
@qm
+qm
p0
q0
� qm
2p0
◆
'�¯�1,¯�2,¯�3
(p, q)
�
�
�
�
�
Cq|�|+1�|⌫1|0
p|¯�1|+|¯�3|�|�|�1�|⌫2|
0
,
= Cq|�0|�|⌫1|
0
p|�1|+|�3|�|�0|�|⌫2|0
.
This establishes (2.46) for (2.52) and therefore for all of the coe�cients. ⇤
Next, we compute derivatives of the collision kernel in (2.26) which will be im-
portant for showing that solutions F± to the relativistic Landau-Maxwell system are
positive.
Lemma 2.3. We compute a sum of first derivatives in q of (2.26) as
(2.53)X
j
@qj
�ij(P, Q) = 2⇤(P, Q)
p0
q0
(P ·Qpi � qi) .
34
This term has a second order singularity at p = q. We further compute a sum of
(2.53) over first derivatives in p as
(2.54)X
i,j
@pi
@qj
�ij(P, Q) = 4P ·Qp
0
q0
�
(P ·Q)2 � 1 �1/2 � 0.
This term has a first order singularity.
This result is quite di↵erent from the classical theory, it is straightforward compute
the derivative of the classical kernel in (2.13) as
X
i,j
@vi
@v0j
�ij(v � v0) = 0.
On the contrary, the proof of Lemma 2.3 is quite technical.
Proof. Throught this proof, we temporarily suspend our use of the Einstein
summation convention. Di↵erentiating (2.26), we have
@qj
�ij(P, Q) ⌘@q
j
⇤(P, Q)
p0
q0
Sij(P, Q)
+⇤(P, Q)
p0
q0
✓
@qj
Sij(P, Q)� qj
q2
0
Sij(P, Q)
◆
.
And
@qj
⇤(P, Q) = 2(P ·Q)�
(P ·Q)2 � 1 �3/2
✓
qj
q0
p0
� pj
◆
�3(P ·Q)3
�
(P ·Q)2 � 1 �5/2
✓
qj
q0
p0
� pj
◆
.
Since (2.2) impliesP
j Sij(P, Q)⇣
qj
q0p
0
� pj
⌘
= 0, we conclude
X
j
@qj
⇤(P, Q)Sij(P, Q)
p0
q0
= 0.
Therefore it remains to evaluate the r.h.s. of
(2.55)X
j
@qj
�ij(P, Q) =⇤(P, Q)
p0
q0
X
j
✓
@qj
Sij(P, Q)� qj
q2
0
Sij(P, Q)
◆
.
35
We take a derivative of Sij in (2.26) as
@qj
Sij = 2 (P ·Q)
✓
qj
q0
p0
� pj
◆
�ij +
✓
qj
q0
p0
� pj
◆
(piqj + qipj)
+ {P ·Q� 1} (pi + �ijpj) + (1 + �ij) (pi � qi)
= 2 (P ·Q)
✓
qj
q0
p0
� pj
◆
�ij +
✓
q2
j
p0
q0
pi + pjqjp
0
q0
qi � pipjqj � qip2
j
◆
+P ·Q (1 + �ij) pi � (1 + �ij) qi.
Next, sum this expression over j to obtain
X
j
@qj
Sij = 2 (P ·Q)
✓
qi
q0
p0
� pi
◆
+p
0
q0
�
|q|2pi + p · qqi
�
�p · qpi � |p|2qi + 4P ·Qpi � 4qi.
We collect terms which are coe�cients of pi and qi respectively
X
j
@qj
Sij = qi
⇢
2 (P ·Q)p
0
q0
+ p · qp0
q0
� |p|2 � 4
�
+pi
⇢
�2P ·Q + |q|2p0
q0
� p · q + 4P ·Q�
= qi
✓
p0
q0
P ·Q� 3
◆
+ pi
✓
3P ·Q� p0
q0
◆
,(2.56)
where the last line follows from plugging |p|2 = p2
0
� 1 = p0
q0
p0
q0� 1 into the first line
and plugging
|q|2p0
q0
= q2
0
p0
q0
� p0
q0
= p0
q0
� p0
q0
into the second line.
Turning to the computation of � 1
q20
P
j qjSij, we plug (2.26) into the following
X
j
qjSij(P, Q) =
�
(P ·Q)2 � 1
qi � (pi � qi) q · (p� q)
+ {(P ·Q)� 1}�
p · qqi + |q|2pi
�
36
We collect terms which are coe�cients of pi and qi respectively to obtain
= qi
�
(P ·Q)2 � 1 + q · (p� q) + p · q {P ·Q� 1}�
+pi
�
�q · (p� q) + |q|2 {P ·Q� 1}�
= qi
�
(P ·Q)2 � 1� |q|2 + p · q (P ·Q)�
+ pi
�
�p · q + |q|2P ·Q�
= qi
�
|p|2q2
0
� p0
q0
p · q�
+ pi
�
|q|2P ·Q� p · q�
= qi
�
p2
0
q2
0
� p0
q0
p · q � q2
0
�
+ pi
�
q2
0
P ·Q� P ·Q� p · q�
= p0
q0
qi
✓
P ·Q� q0
p0
◆
+ q2
0
pi
✓
P ·Q� p0
q0
◆
.
Divide this expression by q2
0
to conclude
X
j
qj
q2
0
Sij = qi
✓
p0
q0
P ·Q� 1
◆
+ pi
✓
P ·Q� p0
q0
◆
.(2.57)
This and (2.56) are very symmetric expressions.
We combine (2.56) and (2.57) to obtain
X
j
✓
@qj
Sij � qj
q2
0
Sij
◆
= qi
✓
p0
q0
P ·Q� 3
◆
+ pi
✓
3P ·Q� p0
q0
◆
�qi
✓
p0
q0
P ·Q� 1
◆
� pi
✓
P ·Q� p0
q0
◆
= 2 (P ·Qpi � qi) .
We note that this term has a first order cancellation at p = q. We plug this last
display into (2.55) to obtain (2.53).
We di↵erentiate (2.53) to obtain
X
i
@pi
X
j
@qj
�ij(P, Q) = 2X
i
@pi
⇤(P, Q)
p0
q0
(P ·Qpi � qi)
+2⇤(P, Q)
p0
q0
X
i
✓
@pi
� pi
p2
0
◆
(P ·Qpi � qi) .(2.58)
And we can write the derivative of ⇤ as
@pi
⇤(P, Q) = �(P ·Q)�
(P ·Q)2 � 1 �5/2
✓
pi
p0
q0
� qi
◆
�
(P ·Q)2 + 2�
.
37
We compute
X
i
✓
pi
p0
q0
� qi
◆
((P ·Q)pi � qi) =X
i
✓
q0
p0
(P ·Q)p2
i �q0
p0
piqi � piqi(P ·Q) + q2
i
◆
=q0
p0
(P ·Q)|p|2 � q0
p0
p · q � p · q(P ·Q) + |q|2.
We further add and subtract q0
p0(P ·Q) to obtain
= p0
q0
(P ·Q)� q0
p0
(P ·Q)� q0
p0
p · q � p · q(P ·Q) + |q|2
= p0
q0
(P ·Q)� q2
0
� p · q(P ·Q) + |q|2
= p0
q0
(P ·Q)� p · q(P ·Q)� 1
= (P ·Q)2 � 1.
We conclude that
(2.59)X
i
@pi
⇤(P, Q)
p0
q0
(P ·Qpi � qi) = �P ·Qp
0
q0
((P ·Q)2 + 2)�
(P ·Q)2 � 1
3/2
.
This term has a third order singularity. We will find that the second term in (2.58)
also has a third order singularity, but there is second order cancellation between the
two terms in (2.58).
We now evaluate the sum in second term in (2.58) as
X
i
✓
@pi
� pi
p2
0
◆
(P ·Qpi � qi) =X
i
✓
P ·Q + pi
✓
piq0
p0
� qi
◆
� P ·Qp2
i
p2
0
+piqi
p2
0
◆
= 3P ·Q + |p|2 q0
p0
� p · q � P ·Q |p|2p2
0
+p · qp2
0
.
We add and subtract q0
p0as well as P ·Q
p20
to obtain
= 3P ·Q� q0
p0
+ p0
q0
� p · q � P ·Q +P ·Q
p2
0
+p · qp2
0
= 3P ·Q� q0
p0
+p
0
q0
p2
0
= 3P ·Q.
Therefore, plugging in (2.26), we obtain
⇤(P, Q)
p0
q0
X
i
✓
@pi
� p2
i
p2
0
◆
(P ·Qpi � qi) = 3(P ·Q)3
p0
q0
�
(P ·Q)2 � 1 �3/2
.
Further plugging this and (2.59) into (2.58) we obtain (2.54). ⇤38
In the following Lemma, we will use Lemma 2.3 to obtain a simplified expression
for part of the collision operator (2.1) which will be used to prove the positvity of our
solutions F± to the relativistic Landau-Maxwell system.
Lemma 2.4. Given a smooth scalar function G(q) which decays rapidly at infinity,
we have
�@pi
Z
R3
�ij(P, Q)@qj
G(q)dq = 4
Z
R3
P ·Qp
0
q0
�
(P ·Q)2 � 1 �1/2
G(q)dq
+(p)G(p),
where (p) = 27/2⇡p0
R ⇡
0
�
1 + |p|2 sin2 ✓��3/2
sin ✓d✓.
Proof. We write out @pi
as in (2.47) to observe
�@pi
Z
�ij(P, Q)@qj
G(q)dq = �Z
⇢
�q0
p0
@qi
+ ⇥ei
�
�ij(P, Q)@qj
G(q)dq
= �Z
⇥ei
�ij(P, Q)@qj
G(q)dq
+
Z
q0
p0
@qi
�ij(P, Q)@qj
G(q)dq.
We split these integrals into |p � q| ✏ and |p � q| > ✏ for ✏ > 0. We note that the
integrals over |p� q| ✏ converge to zero as ✏ # 0. We will eventually send ✏ # 0, so
we focus on the region |p� q| > ✏. We rewrite
q0
p0
@qi
�ij(P, Q)@qj
G(q) = @qj
⇢
q0
p0
@qi
�ij(P, Q)G(q)
�
�@qj
⇢
q0
p0
@qi
�ij(P, Q)
�
G(q).
After an integration by parts, the integrals over |p� q| > ✏ are
=
Z
|p�q|>✏
@qj
�
⇥ei
�ij(P, Q)
G(q)dq
�Z
|p�q|>✏
@qj
⇢
q0
p0
@qi
�ij(P, Q)
�
G(q)dq
+
Z
|p�q|>✏
@qj
⇢
q0
p0
@qi
�ij(P, Q)G(q)
�
dq.
39
By the definition of ⇥ei
in (2.33), this is
=
Z
|p�q|>✏
@qj
@pi
�ij(P, Q)G(q)dq +
Z
|p�q|>✏
@qj
⇢
q0
p0
@qi
�ij(P, Q)G(q)
�
dq.
We plug (2.54) into the first term above to obtain the first term on the r.h.s. of this
Lemma as ✏ # 0. For the second term above, we apply the divergence theorem to
obtain
Z
|p�q|>✏
@qj
⇢
q0
p0
@qi
�ij(P, Q)G(q)
�
dq =
Z
|p�q|=✏
q0
p0
@qi
�ij(P, Q)pj � qj
|p� q|G(q)dS,
where dS is given below. By a Taylor expansion, P ·Q = 1 + O(|p� q|2). Using this
and (2.53) we have
q0
p0
@qi
�ij(P, Q)pj � qj
|p� q| = 2⇤(P, Q)
p2
0
|p� q|+ O(|p� q|�1).
And the integral over |p� q| = ✏ which includes the terms in O(|p� q|�1) goes to zero
as ✏ # 0. We focus on the main part
2p�2
0
Z
|p�q|=✏
⇤(P, Q)|p� q|G(q)dS.
We multiply and divide by p0
q0
+ p · q + 1 to observe that
P ·Q� 1 =|p� q|2 + |p⇥ q|2p
0
q0
+ p · q + 1.
This and (2.26) imply
⇤ = (P ·Q)2
✓
p0
q0
+ p · q + 1
p0
q0
� p · q + 1
◆
3/2
�
|p� q|2 + |p⇥ q|2��3/2
.
We change variables as q ! p� q so that the integrand becomes ⇤|q|G(p� q) and we
define q0
=p
1 + |p� q|2 so that after the change of variables
⇤ = (p0
q0
� |p|2 + p · q)2
✓
p0
q0
+ |p|2 � p · q + 1
p0
q0
� |p|2 + p · q + 1
◆
3/2
�
|q|2 + |p⇥ q|2��3/2
.
We choose the angular integration over |q| = ✏ in such a way that p · q = |p||q| cos ✓
and dS = ✏2 sin ✓d✓d� = ✏2d! with 0 ✓ ⇡, 0 � 2⇡. Note that as ✏ # 0 (on
|q| = ✏)
✏3⇤! 23/2
�
1 + |p|2 sin2 ✓��3/2
p3
0
.
40
Hence, as ✏ # 0,
2p�2
0
Z
|q|=✏
⇤|q|G(p� q)dS = 2p�2
0
Z
S2
✏3⇤G(p� ✏!)d! ! (p)G(p),
with defined in the statement of this Lemma. ⇤
Lemma 2.5. There exists C > 0, such that
(2.60)1
C
�
|rpg|22
+ |g|22
|g|2� C�
|rpg|22
+ |g|22
.
Further, �ij(p) is a smooth function satisfying
(2.61)�
�@��ij(p)
�
� Cp�|�|0
.
Proof. The spectrum of �ij(p), (2.27), consists of a simple eigenvalue �1
(p) > 0
associated with the vector p and a double eigenvalue �2
(p) > 0 associated with p?;
there are constants c1
, c2
> 0 such that, as |p| ! 1, �1
(p) ! c1
, �2
(p) ! c2
. In
Lemou [47] there is a full discussion of these eigenvalues. This is enough to prove
(2.60); see [38] for more details on a similar argument.
We move on to (2.61). We combine (2.27) and (2.45) (with µ(p, q) = J1/2(q)) to
obtain
@��ij(p) = @�
Z
R3
�ij(P, Q)J(q)dq
=X
�1+�2�
Z
R3
⇥�1�(P, Q)J1/2(q)@q�2
J1/2(q)'��1,�2,0(p, q)dq.
By (2.46) then
�
�@��ij(p)
�
� CX
�1+�2�
p|�1|�|�|0
Z
|⇥�1�(P, Q)| J1/2(q)dq.
Recall (2.34), we split this integration into the sets A, B. We plug in the estimate
(2.35) to get
p|�1|�|�|0
Z
A|⇥�1�(P, Q)| J1/2(q)dq Cp|�1|�|�|
0
p�|�1|0
= Cp�|�|0
.
41
On B we have a first order singularity but q is larger than p, in fact we use (2.36) to
get exponential decay in p over this region. With (2.37) we obtainZ
B|⇥�1�(P, Q)| J1/2(q)dq CJ1/16(p)
Z
|p� q|�1 J1/4(q)dq.
We can now deduce (2.61). ⇤
We now write the Landau Operators A, K,� in a new form which will be used
throughout the rest of the paper.
Lemma 2.6. We have the following representations for A, K,� 2 R2, which are
defined in (2.22) and (2.23),
Ag = 2J�1/2@pi
{J1/2�ij(@pj
g +pj
2p0
g)}
= 2@pi
(�ij@pj
g)� 1
2�ij pi
p0
pj
p0
g + @pi
⇢
�ij pj
p0
�
g,(2.62)
Kg = �J(p)�1/2@pi
⇢
J(p)
Z
R3
�ijJ1/2(q)@qj
(g(q) · [1, 1])dq
�
[1, 1](2.63)
�J(p)�1/2@pi
⇢
J(p)
Z
R3
�ijJ1/2(q)qj
2q0
(g(q) · [1, 1])dq
�
[1, 1],
where �ij = �ij(P, Q). Further
(2.64) �(g, h) = [�+
(g, h),��(g, h)],
where
�±(g, h) =
✓
@pi
� pi
2p0
◆
Z
�ijJ1/2(q)@pj
g±(p) (h(q) · [1, 1]) dq,
�✓
@pi
� pi
2p0
◆
Z
�ijJ1/2(q)g±(p)@qj
(h(q) · [1, 1]) dq.
Proof. For (2.62) it su�ces to consider 2J(p)�1/2C(J1/2g±, J):
⌘ 2J(p)�1/2@pi
Z
R3
�ij(P, Q)�
@pj
�
J1/2g±(p)�
J(q)��
J1/2g±(p)�
@qj
J(q)
dq
= 2J(p)�1/2@pi
Z
R3
�ij(P, Q)J(q)J(p)1/2
⇢
@pj
g± +
✓
qj
q0
� pj
2p0
◆
g±
�
dq
= 2J(p)�1/2@pi
⇢
�ij(p)J(p)1/2
✓
@pj
g± +pj
2p0
g±
◆�
.
42
Above, we have used the null space of � in (2.2). Below, we move some derivatives
inside and cancel out one term.
= 2@pi
⇢
�ij(p)
✓
@pj
g± +pj
2p0
g±
◆�
� pi
p0
⇢
�ij(p)
✓
@pj
g± +pj
2p0
g±
◆�
= 2@pi
�
�ij(p)@pj
g±
+ @pi
⇢
�ij(p)pj
p0
�
g± �1
2�ij(p)
pi
p0
pj
p0
g±.
For K simply plug (2.25) with normalized constants into (2.22). For �, we use
the null condition (2.2) to compute J(p)�1/2C(p
Jg+
,p
Jh�)
= J(p)�1/2@pi
Z
�ij(P, Q)J1/2(q)J1/2(p)�
h�(q)@pj
g+
(p)� @qj
h�(q)g+
(p)
dq
+J(p)�1/2@pi
Z
�ij(P, Q)J1/2(q)J1/2(p)
⇢
qj
2q0
� pj
2p0
�
h�(q)g+
(p)dq
= J(p)�1/2@pi
Z
�ij(P, Q)J1/2(q)J1/2(p)�
h�(q)@pj
g+
(p)� @qj
h�(q)g+
(p)
dq
=
✓
@pi
� pi
2p0
◆
Z
�ij(P, Q)J1/2(q)�
h�(q)@pj
g+
(p)� @qj
h�(q)g+
(p)
dq.
Plug four of these type calculations into (2.23) to obtain (2.64). ⇤
We will use these expressions just proven to get the estimates below.
Lemma 2.7. Let |�| > 0. For any small ⌘ > 0, there exists C⌘ > 0 such that
�h@�{Ag}, @�gi � |@�g|2� � ⌘X
|↵||�||@↵g|2� � C⌘|g|2
2
,(2.65)
|h@�{Kg}, @�hi|
8
<
:
⌘X
|¯�||�|
�
�@¯�g�
�
�+ C⌘ |g|
2
9
=
;
|@�h|� .(2.66)
Proof. We will prove (2.65) for a real valued function g to make the notation
less cumbersome, although the result follows trivially for g = [g+
, g�]. We write out
43
the inner product in (2.65) using (2.62) to achieve
h@�{Ag}, @�gi =
Z
R3
@�
✓
@pi
⇢
�ij pi
p0
�
g
◆
@�gdp.
�Z
R3
⇢
1
2@�
✓
�ij pi
p0
pj
p0
g
◆
@�g + 2@�[�ij@pj
g]@pi
@�g
�
dp
= �|@�g|2� +X
↵�
C↵�
Z
R3
@��↵@pi
⇢
�ij pi
p0
�
@↵g@�gdp(2.67)
�X
↵<�
C↵�
Z
R3
2@��↵�ij@↵@p
j
g@pi
@�gdp
�X
↵<�
C↵�
Z
R3
1
2@��↵
✓
�ij pi
p0
pj
p0
◆
@↵g@�gdp
Since ↵ < � in the last two terms below, (2.61) gives us the following estimate�
�
�
�
@��↵@pi
⇢
�ij pi
p0
�
�
�
�
�
+�
�@��↵�ij�
�+
�
�
�
�
@��↵
✓
�ij pi
p0
pj
p0
◆
�
�
�
�
Cp�1
0
We bound the second and fourth terms in (2.67) by
CX
↵�
Z
R3
p�1
0
|@↵g@�g| dp =
Z
|p|m
+
Z
|p|>m
On the unbounded part we use Cauchy-Schwartz
X
↵�
Z
|p|>m
p�1
0
|@↵g@�g| dp C
m|@�g|�
X
↵�
|@↵g|� C
m
X
↵�
|@↵g|2�
On the compact part we use the compact interpolation of Sobolev-spaces
Z
|p|m
X
↵�
|@↵g@�g| dp Z
|p|m
X
↵�
|@↵g|2 + |@�g|2 dp
⌘0X
|↵|=|�|+1
Z
|p|m
|@↵g|2 dp + C⌘0
Z
|p|m
|g|2 dp
⌘X
|↵||�||@↵g|2� + C⌘|g|2
2
For the third term in (2.67), we split into two cases, first suppose |↵| < |�| � 1 and
integrate by parts on @pi
to obtain
X
|↵|<|�|�1
Z
R3
2�
@��↵@pi
�ij@↵@pj
g + @��↵�ij@p
i
@↵@pj
g�
@�gdp
44
We bound this term by
CX
|↵|<|�|
Z
R3
p�1
0
�
�@↵@pj
g@�g�
� dp =
Z
|p|m
+
Z
|p|>m
Z
|p|m
+C
m|@�g|�
X
|↵|<|�||@↵g|�
Z
|p|m
+C
m
X
|↵||�||@↵g|2�
By the compact interpolation of Sobolev spaces
X
|↵|<|�|
Z
|p|m
�
�@↵@pj
g@�g�
� dp ⌘X
|↵||�||@↵g|2� + C⌘|g|2
2
Finally, if |↵| = |�| � 1 for the third term in (2.67), we integrate by parts and use
symmetry
2
Z
R3
@��↵�ij@↵@p
j
g@pi
@�gdp = 2
Z
R3
@��↵�ij@↵@p
j
g@��↵@↵@pi
gdp
= �Z
R3
@2
��↵�ij@↵@p
j
g@↵@pi
gdp.
Because the order of the derivatives on g is now = |�|, we can again use the compact
interpolation of Sobolev spaces and (2.61) to get the same bounds as for the last case
|↵| < |�|� 1 . We obtain
�h@�{Ag}, @�gi � |@�g|2� �✓
⌘ +C
m
◆
X
|↵||�||@↵g|2� � C⌘|g|2
2
,
This completes the estimate (2.65).
We now consider h@�{Kg}, @�hi and (2.66). Recalling (2.63), we use (2.45) with
µ(p, q) =n
@qj
+ qj
2q0
o
(g(q) · [1, 1]), the Leibnitz formula as well as an integration by
parts to express h@�{Kg}, @�hi as
X
ZZ
⇥↵1�ij(P, Q)
p
J(p)J(q)@q↵2
⇢
@qj
+qj
2q0
�
(g(q) · [1, 1])
⇥@�1(h(p) · [1, 1])'�,�1↵1,↵2,0(p, q)dqdp,
where the sum is ↵1
+↵2
�, |�| |�1
| |�|+1. And '�,�1↵1,↵2,0(p, q) is a collection of
the inessential terms, it satisfies the decay estimate (2.46) independent of the value
45
of �1
. Therefore we can further express h@�{Kg}, @�hi as
X
ZZ
µ�1�2(p, q)J(p)1/4J(q)1/4@�1(h(p) · [1, 1])@q�2
(g(q) · [1, 1])dqdp
where the sum is over |�2
| |�| + 1, |�| |�1
| |�| + 1. And, using (2.35) and
(2.37), we see that µ�1�2(p, q) is a collection of L2 functions. Therefore, as in (2.80),
we split h@�{Kg}, @�hi to get
X
ZZ
ij(p, q)J(p)1/4J(q)1/4@�1(h(p) · [1, 1])@q�2
(g(q) · [1, 1])dqdp
+X
ZZ
{µ�1�2 � ij} J(p)1/4J(q)1/4@�1(h(p) · [1, 1])@q�2
(g(q) · [1, 1])dqdp.
In the same fashion as J2
in (2.80), for any m > 0, we estimate the second term above
by
C
m|@�h|�
X
|¯�||�|
�
�@¯�g�
�
�.
Since ij(p, q) is a smooth function with compact support, we integrate by parts over
q repeatedly to bound the first term by
X
�
�
�
�
ZZ
@�1{ ij(p, q)J(q)1/4}J(p)1/4(g(q) · [1, 1])@�1(h(p) · [1, 1])dqdp
�
�
�
�
C(m)X
ZZ
J(q)1/8J(p)1/4 |(g(q) · [1, 1])@�1(h(p) · [1, 1])| dqdp
C(m) |g|2
X
|@�1h|2
C(m) |g|2
|@�h|� .
where the final sum above is over |�| |�1
| |�| + 1. And we have used (2.60) to
get the last inequality. We conclude our lemma by choosing m large. ⇤
We now estimate the nonlinear term �(f, g):
Theorem 2.4. Let |�|+ |�| N, then
�
�h@���(f, g), @�
�hi�
� CX
{|@�1�3
f |2
|@���1�2
g|� + |@�1�3
f |�|@���1�2
g|2
}|@��h|�,(2.68)
where the summation is over �1
�, �2
+ �3
�. Further
�
�(@���(f, g), @�
�h)�
� Ck@��hk� {|||g|||�|||f |||+ |||f |||�|||g|||} .
46
Proof. Notice @��(f, g) =P
�1� C�1� �(@�1f, @���1g); thus it su�ces to only
consider the p derivatives. From (2.64), using (2.45), we can write @��(f, g) as
@pi
Z
⇥�1�ij(P, Q)
p
J(q)@q�2@�3
�
@pj
fl(p)gk(q)
'��1,�2,�3
(p, q)dq
�@pi
Z
⇥�1�ij(P, Q)
p
J(q)@q�2@�3
�
fl(p)@qj
gk(q)
'��1,�2,�3
(p, q)dq
+
Z
⇥�1�ij(P, Q)
p
J(q)@q�2@�3
⇢
fl(p)@qj
gk(q)pi
2p0
�
'��1,�2,�3
(p, q)dq
�Z
⇥�1�ij(P, Q)
p
J(q)@q�2@�3
⇢
@pj
fl(p)gk(q)pi
2p0
�
'��1,�2,�3
(p, q)dq.
Above, we implicitly sum over i, j 2 {1, 2, 3}, �1
+ �2
+ �3
� and k 2 {+,�}. And
l 2 {+,�}. Recall '��1,�2,�3
(p, q) from Theorem 2.3. Further (2.46) implies
�
�
�
'��1,�2,�3
(p, q)�
�
�
Cq|�|0
.
We use the above two displays and integrate by parts over @pi
in the first two terms,
to bound |h@��(f, g), @�hi| above by
C
ZZ
J1/4(q)�
�⇥�1�ij(P, Q)@q
�2gk(q)@�3@p
j
fl(p)@pi
@�hl(p)�
� dqdp(2.69)
+C
ZZ
J1/4(q)�
�⇥�1�ij(P, Q)@q
�2@q
j
gk(q)@�3fl(p)@pi
@�hl(p)�
� dqdp(2.70)
+C
ZZ
J1/4(q)�
�⇥�1�ij(P, Q)@q
�2@q
j
gk(q)@�3fl(p)@�hl(p)�
� dqdp
+C
ZZ
J1/4(q)�
�⇥�1�ij(P, Q)@q
�2gk(q)@�3@p
j
fl(p)@�hl(p)�
� dqdp.
In the above expressions, we add the summation over l 2 {+,�}. It su�ces to
estimate (2.69) and (2.70); the other two terms are similar. While estimating these
terms we will repeatedly use the equivalence of the norms in (2.60) without mention.
We start with (2.69).
47
We next split (2.69) into the two regions, A and B, defined in (2.34). We then
use (2.37) and the Cauchy-Schwartz inequality to estimate (2.69) over B
ZZ
BJ1/4(q)
�
�⇥�1�ij(P, Q)@q
�2gk(q)@�3@p
j
fl(p)@pi
@�hl(p)�
� dqdp
C
ZZ
B|p� q|�1J1/8(q)
�
�@q�2
gk(q)@�3@pj
fl(p)@pi
@�hl(p)�
� dqdp
C|@�h|�
Z
R3
�
�@�3@pj
fl(p)�
�
2
✓
Z
B|p� q|�1J1/8(q)
�
�@q�2
gk(q)�
� dq
◆
2
dp
!
1/2
C|@�h|�|@�2g|2✓
Z
R3
�
�@�3@pj
fl(p)�
�
2
✓
Z
B|p� q|�2J1/4(q)dq
◆
dp
◆
1/2
C|@�h|�|@�2g|2|@�3f |�,
where use (2.36) to say thatR
B |p� q|�2J1/4(q)dq C. This completes the estimate
of (2.69) over B.
We estimate (2.69) over A using (2.35)
ZZ
AJ1/4(q)
�
�⇥�1�ij(P, Q)@q
�2gk(q)@�3@p
j
fl(p)@pi
@�hl(p)�
� dqdp
C
ZZ
J1/8(q)�
�@q�2
gk(q)@�3@pj
fl(p)@pi
@�hl(p)�
� dqdp
C|@�h|�|@�2g|2|@�3f |�.
This completes the estimate for (2.69).
Note that the di↵erence between (2.70) and (2.69) is that @qj
hits g in (2.70)
whereas @pj
hit f in (2.69). This di↵erence means we will need to use the norm | · |�to estimate g but we are able to use the smaller norm | · |
2
to estimate f . This is in
contrast to the estimate for (2.69) where the opposite situation held. The main point
is that | · |� includes first order p derivatives.
Using the same splitting over A and B as well as the same type calculation, (2.70)
is bounded by |@�h|�|@�2g|�|@�3f |2. This completes the estimate for (2.70).
The proof of (2.69) follows from the Sobolev embedding: H2(T3) ⇢ L1(T3).
Without loss of generality, assume |�1
| N/2. This Sobolev embedding grants us
48
that
✓
supx|@�1
�2f(x)|
2
◆
|@���1�3
g(x)|� +
✓
supx|@�1
�2f(x)|�
◆
|@���1�3
g(x)|2
⇣
X
k@��2
fk⌘
|@���1�3
g(x)|� +
X
k@��2
fk�
!
|@���1�3
g(x)|2
,
where summation is over |�| |�1
| + 2 N since N � 4. We conclude (2.69) by
integrating (2.68) further over T3. ⇤
We next prove the important estimates which are needed to prove Theorem 2.2
in Section 2.5.
Theorem 2.5. Let |�| N . let g(x, p) be a smooth vector valued L2(T3
x⇥R3
p; R2)
function and h(p) a smooth vector valued L2(R3
p; R2) function, we have
(2.71) kh@��(g, g), hik CX
|�|2
|@�h|2
X
|�|N
k@�gkX
|�|N
k@�gk�
Moreover,
(2.72) khL@�g, hik Ck@�gkX
|�|2
|@�h|2
Proof. We begin with the linear term. By Lemma 2.6, (2.21) and two integra-
tions by parts hL@�g, hi is given by
Z
⇢
�@�g · @pj
�
@pi
h(p)2�ij�
+1
2�ij pi
p0
pj
p0
@�g · h(p)
�
dp
�Z
⇢
@pi
⇢
�ij pi
p0
�
@�g · h(p)
�
dp
�ZZ
qj
2q0
�ij(P, Q)p
J(q)J(p)@�gl(q)@pi
�
J(p)�1/2hk(p)
dqdp,
+
ZZ
@qj
n
�ij(P, Q)p
J(q)o
J(p)@�gl(q)@pi
�
J(p)�1/2hk(p)
dqdp
49
we implicitly sum over i, j 2 {1, 2, 3} and k, l 2 {+,�}. Using cauchy’s inequality,
khL@�g, hik2 is bounded by
2
Z
✓
Z
⇢
�@�g · @pj
�
@pi
h(p)2�ij�
+1
2�ij pi
p0
pj
p0
@�g · h(p)
�
dp
◆
2
dx
2
Z
✓
Z
⇢
@pi
⇢
�ij pi
p0
�
@�g · h(p)
�
dp
◆
2
dx(2.73)
+2
Z
✓
ZZ
qj
2q0
�ij(P, Q)p
J(q)J(p)@�gl(q)@pi
�
J(p)�1/2hk(p)
dqdp
◆
2
dx
plus
2
Z
✓
ZZ
@qj
n
�ij(P, Q)p
J(q)o
J(p)@�gl(q)@pi
�
J(p)�1/2hk(p)
dqdp.
◆
2
dx(2.74)
We have split (2.73) and (2.74) because we will estimate each one separately.
By the Cauchy-Schwartz inequality as well as (2.61), and that h(p) is not a func-
tion of x, the first and second lines of (2.73) are bounded by
CX
|�|2
|@�h|22
Z
|@�g(x)|22
dx = Ck@�gk2
X
|�|2
|@�h|22
This completes (2.72) for the first and second lines of (2.73).
By the Cauchy-Schwartz inequality over dp the third line of (2.73) is bounded by
CX
|�|1
|@�h|22
Z
Z
J1/2(q) |@�gl(q)|⇢
Z
�ij(P, Q)2J1/2(p)dp
�
1/2
dq
!
2
dx
Recall again the splitting in (2.34), apply (2.35) and (2.37) (with ↵1
= 0) to obtainZ
�ij(P, Q)2J1/2(p)dp =
Z
A+
Z
B
Cq12
0
+ q14
0
Z
B|p� q|�2J1/4(p)dp
Cq14
0
Using the Cauchy-Schwartz inequality again, this implies that the third line of (2.73)
is bounded by Ck@�gk2
P
|�|1
|@�h|22
.
To establish (2.72) it remains to estimate (2.74). From (2.33), we write
@qj
= �p0
q0
@pj
+
✓
@qj
+p
0
q0
@pj
◆
= �p0
q0
@pj
+ ⇥ej
(q, p)
50
where ej is an element of the standard basis in R3. For the rest of this proof, we write
⇥ej
= ⇥ej
(q, p) for notational simplicity (although the reader should note that it is
the opposite of the shorthand we were using previously). Further,
@qj
�
�ij(P, Q)J1/2(q)
= ��ij(P, Q)qj
2q0
p
J(q)
�p
J(q)p
0
q0
@pj
�ij(P, Q) +p
J(q)⇥ej
�ij(P, Q).
We plug the above into (2.74) and integrate by parts for the middle term in (2.74) to
bound (2.74) by
C
Z
✓
ZZ
p
J(q)p
J(p)�
��ij(P, Q)@�gl(q)�
� |@�hk(p)| dqdp
◆
2
dx(2.75)
+C
Z
✓
ZZ
p
J(q)p
J(p)�
�⇥ej
�ij(P, Q)@�gl(q)�
� |@�hk(p)| dqdp
◆
2
dx.
Above, we add the implicit summation over |�| 2. By the same estimates as for
(2.73), the first term in (2.75) is Ck@�gk2
P
|�|2
|@�h|22
.
To establish (2.72), it remains to estimate the second term in (2.75). To this end,
we note that ⇥ej
(q, p)P ·Q = 0. Further, ⇥ej
(q, p)�ij(P, Q) satisfies the estimates
|⇥ej
(q, p)�(P, Q)| Cq6
0
q�1
0
on A,
|⇥ej
(q, p)�(P, Q)| Cq7
0
q�1
0
|p� q|�1 on B,
where we recall the sets from (2.34). This can be shown directly, by repeating
the proof of (2.35) and (2.37) using the operator ⇥ej
(q, p) in place of the opera-
tor ⇥ej
(p, q). Plugging these estimates into the second term of (2.75), we can show
that it is bounded by Ck@�gk2
P
|�|2
|@�h|22
using the same estimates as used for
(2.73). This completes the estimate (2.72).
We turn to the estimate for the non-linear term (2.71). This estimate employs
the same idea as for the linear term (2.72), i.e. to move the momentum derivatives
around so that we can get an upper bound in terms of at least one k · k norm.
51
The inner product h@��(g, g), hi, using (2.64) and an integration by parts, is equal
to
�C�1�
ZZ
�ijp
J(q)@pj
@�1gl(p)@���1gk(q)
✓
@pi
+pi
2p0
◆
hl(p)dqdp,
+C�1�
ZZ
�ijp
J(q)@�1gl(p)@qj
@���1gk(q)
✓
@pi
+pi
2p0
◆
hl(p)dqdp.
Above we implicitly sum is over i, j 2 {1, 2, 3}, �1
� and l, k 2 {+,�}.
Assume, without loss of generality, that |�1
| N/2 (and N � 4). We then
integrate by parts with respect to @pj
for the first term above. After this integration by
parts, we obtain a term like @pj
�ij. For this term we write it as @pj
�ij = � q0
p0@q
j
�ij +
⇥ej
(p, q)�ij, where we used the notation (2.33). We then integrate by parts again for
this term with respect to @qj
. The result is bounded above by
C
ZZ
J1/4(q)�
�⇥ej
(p, q)�ij�
�
�
�@�1gl(p)@���1gk(q)�
� |@�hl(p)| dqdp,
+C
ZZ
p
J(q)�
��ij(P, Q)�
�
�
�@�1gl(p)@q�1@���1gk(q)
�
� |@�hl(p)| dqdp,
where we sum over everything from the last display as well as over |�1
| 1 and
|�| 2. The main point is that we took @pj
o↵ of the function on which we want
to have an L2 estimate (the function with less spatial derivatives). We use the same
procedure used to estimate (2.69) to obtain the upper bound (for both terms above)
CX
|�|2
|@�h|2
X
|�1|N/2
|@�1g|2
|@���1g|�
Therefore |h@��(g, g), hi|2 CP
|�|2
|@�h|22
P
|�1|N/2
|@�1g|22
|@���1g|2�. Further inte-
grating over T3 we get
kh@��(g, g), hik2 CX
|�|2
|@�h|22
X
|�1|N/2
Z
|@�1g|22
|@���1g|2�dx.
We establish (2.71) by using the Sobolev embedding: H2(T3) ⇢ L1(T3). ⇤
We end this section with a proof that L is coercive away from it’s null space N .
52
Lemma 2.8. For any m > 1, there is 0 < C(m) < 1 such that
|h@pi
⇢
�ij pj
p0
�
g, hi|+ |hKg, hi|
C
m|g|� |h|� + C(m)
⇢
Z
|p|C(m)
|g|2dp
�
1/2
⇢
Z
|p|C(m)
|h|2dp
�
1/2
.(2.76)
Moreover, there is � > 0, such that
(2.77) hLg, gi � �| (I�P) g|2�.
Proof. We first prove (2.76). We split
(2.78)
Z
@pi
⇢
�ij pj
p0
�
{g+
h+
+ g�h�} dp =
Z
{|p|m}+
Z
{|p|�m}.
By (2.61)�
�
�
�
@pi
⇢
�ij pj
p0
�
�
�
�
�
Cp�1
0
.
So the first integral in (2.78) is bounded by the right hand side of (2.76). From the
Cauchy-Schwartz inequality and (2.60) we obtain
(2.79)
Z
{|p|�m} C
m
Z
|g||h|dp C
m|g|� |h|� .
Consider the linear operator K in (2.63). After an integration by parts we can write
hKg, hi =X
ZZ
�ijJ1/4(p)J1/4(q) ↵1↵2(p, q)@↵1gk(q)@↵2hl(p)dqdp,
where �ij = �ij(P, Q) and the sum is over i, j 2 {1, 2, 3}, |↵1
| 1, |↵2
| 1 and
k, l 2 {+,�}. Also, ↵1↵2(p, q) is a collection of smooth functions, in which we collect
all the inessential terms, that satisfies
|r ↵1↵2(p, q)|+ | ↵1↵2(p, q)| CJ1/8(p)J1/8(q).
From (2.26) as well as Proposition 2.1,
�ij(P, Q)J1/4(p)J1/4(q) 2 L2(R3 ⇥ R3).
Therefore, for any given m > 0, we can choose a C1c function ij(p, q) such that
||�ijJ1/4(p)J1/4(q)� ij||L2(R3
p
⇥R3q
)
1
m,
supp{ ij} ⇢ {|p|+ |q| C(m)}, C(m) < 1.
53
We split
�ijJ1/4(p)J1/4(q) = ij + {�ijJ1/4(p)J1/4(q)� ij}
and
(2.80) hKg, hi = J1
(g, h) + J2
(g, h),
with
J1
=X
ZZ
ij(p, q) ↵1↵2(p, q)@↵1gk(q)@↵2hl(p)dqdp,
J2
=X
ZZ
{�ijJ1/4(p)J1/4(q)� ij} ↵1↵2(p, q)@↵1gk(q)@↵2hl(p)dqdp.
The second term J2
is bounded in absolute value by
||�ijJ1/4(p)J1/4(q)� ij||L2(R3
p
⇥R3q
)
|| ↵1↵2@↵1gk(q)@↵2hl(p)||L2(R3
p
⇥R3q
)
C
m
�
�J1/8@↵1gk
�
�
2
�
�J1/8@↵2hl
�
�
2
C
m|g|� |h|� ,
where we have used the equivalence of the norms (2.60). For J1
, an integration by
parts over p and q yields
J1
=X
(�1)↵1+↵2
ZZ
@↵2@q↵1{ ij(p, q) ↵1↵2(p, q)} gk(q)hl(p)dqdp
C|| ij||C2
⇢
Z
|p|C(m)
|g|2dp
�
1/2
⇢
Z
|p|C(m)
|h|2dp
�
1/2
.(2.81)
This concludes (2.76).
We use the method of contradiction to prove (2.77). The converse grants us a
sequence of normalized functions gn(p) = [gn+
(p), gn�(p)] such that |gn|� ⌘ 1 and
Z
R3
gnJ1/2dp =
Z
R3
pjgnJ1/2dp =
Z
R3
gnp0
J1/2dp = 0,(2.82)
hLgn, gni = �hAgn, gni � hKgn, gni 1/n.(2.83)
We denote the weak limit, with respect to the inner product h·, ·i�, of gn (up to a
subsequence) by g0. Lower semi-continuity of the weak limit implies |g0|� 1. From
(2.62), (2.63) and (2.21) we have
hLgn, gni = |gn|2� � h@pi
⇢
�ij pj
p0
�
gn, gni � hKgn, gni.
54
We claim that
limn!1
h@pi
⇢
�ij pj
p0
�
gn, gni = h@pi
⇢
�ij pj
p0
�
g0, g0i, limn!1
hKgn, gni ! hKg0, g0i.
For any given m > 0, since @pi
gn are bounded in L2{|p| m} from |gn|� = 1 and
(2.60), the Rellich theorem implies
Z
{|p|m}@p
i
⇢
�ij pj
p0
�
(gn)2 !Z
{|p|m}@p
i
⇢
�ij pj
p0
�
(g0)2.
On the other hand, by (2.79) with g = h = gn, the integral over {|p| � m} is
bounded by O(1/m). By first choosing m su�ciently large and then sending n !1,
we conclude h@pi
{�ijpj/p0
}gn, gni ! h@pi
{�ijpj/p0
}g0, g0i.
We split hKgn, gni into J1
and J2
as in (2.80), then J2
(gn, gn) Cm
. In the same
manner as for (2.81), we obtain
�
�J1
(gn, gn)� J1
(g0, g0)�
� C(m)
⇢
Z
|p|C(m)
|gn � g0|2dp
�
1/2
Then the Rellich theorem implies, up to a subsequence, J1
(gn, gn) ! J1
(g0, g0). Again
by first choosing m large and then letting n ! 1, we conclude that hKgn, gni !
hKg0
, g0
i.
Letting n !1 in (2.83), we have shown that
0 = 1� h@pi
{�ijpj/p0
}g0, g0i � hKg0, g0i.
Equivalently
0 =�
1� |g0|2��
+ hLg0, g0i.
Since both terms are non-negative, |g0|2� = 1 and hLg0, g0i = 0. By Lemma 2.1,
g0 = Pg0. On the other hand, letting n !1 in (2.82) we deduce that g0 = (I�P) g0
or g0 ⌘ 0; this contradicts |g0|2� = 1. ⇤
2.4. Local Solutions
We now construct a unique local-in time solution to the relativistic Landau-
Maxwell system with normalized constants (2.29) and (2.30), with constraint (2.31).
55
Theorem 2.6. There exist M0
> 0 and T ⇤ > 0 such that if T ⇤ M0
/2 and
E(0) M0
/2,
then there exists a unique solution [f(t, x, p), E(t, x), B(t, x)] to the relativistic Landau-
Maxwell system (2.29) and (2.30) with constraint (2.31) in [0, T ⇤) ⇥ T3 ⇥ R3 such
that
sup0tT ⇤
E(t) M0
.
The high order energy norm E(t) is continuous over [0, T ⇤). If
F0
(x, p) = J + J1/2f0
� 0,
then F (t, x, p) = J + J1/2f(t, x, p) � 0. Furthermore, the conservation laws (2.10),
(2.11), (2.12) hold for all 0 < t < T ⇤ if they are valid initially at t = 0.
We consider the following iterating sequence (n � 0) for solving the relativis-
tic Landau-Maxwell system for the perturbation (2.29) with normalized constants
(Remark 2.1):
⇢
@t +p
p0
·rx + ⇠
✓
En +p
p0
⇥Bn
◆
·rp � A� ⇠
2
✓
En · p
p0
◆�
fn+1
= ⇠1
⇢
En · p
p0
�pJ + Kfn + �(fn+1, fn)(2.84)
+p
J�
fn+1 � fn�
@pi
Z
R3
�ij(P, Q)@qi
⇣
p
J(q)fn(q) · [1, 1]⌘
dq
+p
J�
fn+1 � fn�
@pi
Z
R3
�ij(P, Q)@qi
J(q)dq,
@tEn �rx ⇥Bn = �J n = �
Z
R3
p
p0
pJ{fn
+
� fn�}dp,
@tBn +rx ⇥ En = 0,
rx · En = ⇢n =
Z
R3
pJ{fn
+
� fn�}dp, rx ·Bn = 0.
Above ⇠1
= [1,�1], and the 2⇥ 2 matrix ⇠ is diag(1,�1). We start the iteration with
f 0(t, x, p) = [f 0
+
(t, x, p), f0
�(t, x, p)] ⌘ [f0,+(x, p), f
0,�(x, p)].
56
Then solve for [E0(t, x), B0(t, x)] through the Maxwell system with initial datum
[E0
(x), B0
(x)]. We then iteratively solve for
fn+1(t, x, p) = [fn+1
+
(t, x, p), fn+1
� (t, x, p)], En+1(t, x), Bn+1(t, x)
with initial datum [f0,±(x, p), E
0
(x), B0
(x)].
It is standard from the linear theory to verify that the sequence [fn, En, Bn] is
well-defined for all n � 0. Our goal is to get an uniform in n estimate for the energy
En(t) ⌘ E(fn, En, Bn)(t).
Lemma 2.9. There exists M0
> 0 and T ⇤ > 0 such that if T ⇤ M02
and
E(0) M0
/2
then sup0tT ⇤ En(t) M
0
implies sup0tT ⇤ En+1
(t) M0
.
Proof. Assume |�|+ |�| N and take @�� derivatives of (2.84), we obtain:
⇢
@t +p
p0
·rx + ⇠
✓
En +p
p0
⇥Bn
◆
·rp
�
@��fn+1
�@�{A@�fn+1}� ⇠1
@�En · @�
⇢
p
p0
J1/2
�
= �X
�1 6=0
C�1� @�1
✓
p
p0
◆
·rx@����1
fn+1(2.85)
+X
C�1�
⇠
2
⇢
@�1En · @�1
✓
p
p0
◆�
@���1���1
fn+1
�⇠X
�1 6=0
C�1� @
�1En ·rp@���1� fn+1
+⇠X
(�1,�1) 6=(0,0)
C�1� C�1
� @�1
✓
p
p0
◆
⇥ @�1Bn ·rp@���1���1
fn+1
+@�{K@�fn}+ @���(fn+1, fn)
+X
C�1� C�1
� @���1
npJ@���1
�
fn+1 � fn�
o
⇥@�1@pi
Z
R3
�ij(P, Q)@qi
⇣
p
J(q)@�1fn(q) · [1, 1]⌘
dq
+X
C�1� @���1
npJ@�
�
fn+1 � fn�
o
@�1@pi
Z
R3
�ij(P, Q)@qi
J(q)dq.
57
We take the inner product of (2.85) with @��fn+1 over T3⇥R3 and estimate this inner
product term by term.
Using (2.65), the inner product of the first two terms on the l.h.s of (2.85) are
bounded from below by
1
2
d
dt||@�
�fn+1(t)||2 + ||@��fn+1(t)||2� � ⌘|||fn+1(t)|||2� � C⌘k@�fn+1(t)k2.
For the third term on l.h.s. of (2.85) we separate two cases. If � 6= 0, its inner product
is bounded by
(2.86)�
�
�
⇣
@�En · @�{pp
J/p0
}⇠1
, @��fn+1
⌘
�
�
�
C||@�En|| |||fn+1|||.
If � = 0, we have a pure temporal and spatial derivative @� = @�0
t @�1
x1@�2
x2@�3
x3. We first
split this term as
�@�En ·✓
p
p0
J1/2
◆
⇠1
⌘ �@�En+1 ·✓
p
p0
J1/2
◆
⇠1
�{@�En � @�En+1} ·✓
p
p0
J1/2
◆
⇠1
.(2.87)
From the Maxwell system (2.30) and an integration by parts the inner product of the
first part is
�⇣
@�En+1 · {pp
J/p0
}⇠1
, @�fn+1
⌘
= �ZZ
@�En+1 ·✓
p
p0
pJ
◆
{@�fn+1
+
� @�fn+1
� }dpdx
= �Z
@�En+1 · @�J n+1dx(2.88)
=1
2
d
dt
�
||@�En+1(t)||2 + ||@�Bn+1(t)||2�
.
And the inner product of second part in (2.87) is bounded by
C{|||En|||+ |||En+1|||}|||fn+1|||.
We now turn to the r.h.s. of (2.85). The first inner product is bounded by (|�1
| � 1)
C|||fn+1|||2. The second, third and fourth inner products on r.h.s. of (2.85) can be
58
bounded by a collection of terms of the same form
CX
Z
T3
{|@�1En|+ |@�1Bn|}✓
Z
R3
|@���1���1
fn+1@��fn+1|dp
◆
dx(2.89)
+CX
(�1,�1) 6=(0,0)
Z
T3
{|@�1En|+ |@�1Bn|}✓
Z
R3
|rp@���1���1
fn+1@��fn+1|dp
◆
dx
where the sums are over �1
�, and �1
�. From the Sobolev embedding H2(T3) ⇢
L1(T3) we have
supx
⇢
Z
R3
|g(x, q)|2dq
�
Z
R3
supx|g(x, q)|2dq C
X
|�|2
||@�g||2.(2.90)
We take the L1 norm in x of the one of first two factors in (2.89) depending on
whether |�1
| N/2 (take the first term) or |�1
| > N/2 (take the second term). Since
N � 4, by (2.90) and (2.60) we can majorize (2.89) by
(2.91) C{|||En|||+ |||Bn|||}|||fn+1|||2 C{|||En|||+ |||Bn|||}|||fn+1|||2�.
We take (2.66), use Cauchy’s inequality with ⌘ and integrate over T3 to obtain
�
@�[K@�fn], @��fn+1
�
⌘|||fn|||2� + ⌘�
�@��fn+1
�
�
2
�+ C⌘ k@�fnk2 .
For the nonlinear term we use Theorem 2.4 to obtain
(@���(fn
, fn+1), @��fn+1) C|||fn(t)||||||fn+1(t)|||�||@�
�fn+1(t)||�
+C|||fn(t)|||�|||fn+1(t)|||||@��fn+1(t)||�,
We turn our attention to the inner product of the second to last term in (2.85). We
integrate by parts over @pi
and apply Theorem 2.3 to the dq integral di↵erentiated by
@�1 . Then this term is bounded by
Z
�
�⇥¯�1�ij(P, Q)
�
� J1/4(q)�
�
�
@↵2@�1¯�2
fnk (q)
�
�
�
�
�
�
@↵2
⇣
@���1¯�3
fml (p)@�
�fn+1
l (p)⌘
�
�
�
dpdqdx,
where we sum over m 2 {n, n + 1}, �1
+ �1
+ �3
�, i, j 2 {1, 2, 3}, k, l 2 {+,�},
|↵1
| 1 |↵2
| 1 and �1
�. We remark that a few of these sum’s are over estimates
used to simplify the presentation. This term is always of the form of one of the four
59
terms in (2.69)-(2.70) up to the location of one p derivative. Therefore, as in the
proof of Theorem 2.4, this term is bounded above by
C�
|||fn+1|||�|||fn|||�|||fn|||+ |||fn|||2�|||fn+1|||�
+C�
|||fn+1|||2�|||fn|||+ |||fn+1|||�|||fn|||�|||fn+1|||�
.
where the sum is over |�i|+ |�i| N , �1
+ �2
�. For the inner product of the last
term in (2.85). The null space in (2.2) implies
Z
R3
�ij(P, Q)@qi
J(q)dq =
Z
R3
�ij(P, Q)qi
q0
J(q)dq = �ij pi
p0
.
Therefore (2.61) applies to the derivatives of the dq integral. Therefore, the inner
product of the last term in (2.85) is bounded by
Z
�
�
�
@�¯�fm
k (p)@��fn+1
k (p)�
�
�
dpdx,
where we sum over m 2 {n, n + 1}, |�| |�| and k 2 {+,�}. This term is bounded
above by
C�
|||fn+1|||2 + |||fn|||(t)|||fn+1|||�
C|||fn+1|||2 + C|||fn|||2.
By collecting all the above estimates, we obtain the following bound for our iter-
ation
1
2
d
dt
�
||@��fn+1(t)||2 + ||@�En+1(t)||2 + ||@�Bn+1(t)||2
�
+ ||@��fn+1(t)||2�
⌘|||fn+1(t)|||2� + C⌘k@�fn+1(t)k2 + C{|||En|||+ |||En+1|||}|||fn+1|||
+C|||fn+1(t)|||2 + C{|||En|||+ |||Bn|||}|||fn+1|||2�
+⌘|||fn|||2� + ⌘�
�@��fn+1
�
�
2
�+ C⌘ k@�fnk2
+C�
|||fn+1|||�|||fn||| + |||fn|||�|||fn+1|||
|||fn+1|||�
+C�
|||fn+1|||�|||fn|||�|||fn|||+ |||fn|||2�|||fn+1|||�
+C|||fn+1|||2 + C|||fn|||2.
60
Summing over |�|+ |�| N and choosing ⌘ 1
4
we have
E 0n+1
(t) C{En+1
(t) + En(t) + E1/2
n (t)|||fn+1|||2�(t)(2.92)
+E1/2
n (t)|||fn|||�(t)|||fn+1|||�(t) + CE1/2
n+1
(t)|||fn|||2�(t)
+1
4C|||fn|||2� + |||fn|||�(t) · |||fn+1|||(t) · |||fn+1|||�(t)}.
By the induction assumption, we have
1
2|||fn|||2(t) + |||En|||2(t) + |||Bn|||2(t) +
Z t
0
|||fn|||2�(s)ds
= En(t) sup0st
En(s) M0
.
Therefore,
Z t
0
|||fn|||�(s) · |||fn+1|||(s) · |||fn+1|||�(s)ds
sup0st
|||fn+1|||(s)⇢
Z t
0
|||fn|||2�(s)
�
1/2
⇢
Z t
0
|||fn+1|||2�(s)
�
1/2
p
M0
sup0st
En+1
(s).
Upon further integrating (2.92) over [0, t] we deduce
En+1
(t) En+1
(0) + C
✓
t sup0st
En+1
(s) + M0
t +p
M0
En+1
(t)
◆
+M
0
4+ CM
0
sup0st
E1/2
n+1
(s) + Cp
M0
sup0st
En+1
(s),
and we will use the inequality
M0
sup0st
En+1
(s) M3/2
0
+p
M0
sup0st
En+1
(s).
From the initial conditions (n � 0)
fn+1
0
⌘ fn+1(0, x, p) = f0
(x, p)
En+1
0
⌘ En+1(0, x) = E0
(x)
Bn+1
0
⌘ Bn+1(0, x) = B0
(x),
61
we deduce that
@��fn+1
0
= @��f
0
, @�En+1
0
= @�E0
, @�Bn+1
0
= @�B0
by a simple induction over the number of temporal derivatives, where the temporal
derivatives are defined naturally through (2.84). Hence
En+1
(0) = En+1
([fn+1
0
, En+1
0
, Bn+1
0
]) ⌘ E([f0
, E0
, B0
]) M0
/2.
It follows that for t T ⇤,
(1� CT ⇤ � CM1/2
0
) sup0tT ⇤
En+1
(t) En+1
(0) + CM0
T ⇤ + CM3/2
0
+M
0
4
3
4M
0
+ CM0
⇣
T ⇤ +p
M0
⌘
.
We therefore conclude Lemma 2.9 if T ⇤ M02
and M0
is small. ⇤
In order to complete the proof of Theorem 2.6, we take n ! 1, and obtain a
solution f from Lemma 2.9. Now for uniqueness, we assume that there is another
solution [g, Eg, Bg], such that sup0sT ⇤ E(g(s)) M
0
with f(0, x, p) = g(0, x, p),
Ef (0, x) = Eg(0, x) and Bf (0, x) = Bg(0, x). The di↵erence [f � g, Ef �Eg, Bf �Bg]
satisfies
⇢
@t +p
p0
·rx + ⇠
✓
Ef +p
p0
⇥Bf
◆
·rp � A
�
(f � g)� (Ef � Eg) ·p
p0
pJ⇠
1
= �⇠⇢
Ef � Eg +p
p0
⇥ (Bf �Bg)
�
rpg + K(f � g)(2.93)
+⇠
⇢
Ef ·p
p0
�
(f � g) + ⇠
⇢
(Ef � Eg) ·p
p0
�
g + �(f � g, f) + �(g, f � g);
@t(Ef � Eg)�rx ⇥ (Bf �Bg) = �Z
p
p0
pJ{(f � g) · ⇠
1
} ,
rx · (Ef � Eg) =
Z pJ{(f � g) · ⇠
1
},
@t(Bf �Bg) +rx ⇥ (Ef � Eg) = 0, rx · (Bf �Bg) = 0.
62
By using the Cauchy-Schwarz inequality in the p�integration, and applying (2.90)
for supx
R
|rpg|2dp, we deduce (for N � 4)
�
�
�
�
✓
u{Ef � Eg +p
p0
⇥ (Bf �Bg)} ·rpg, f � g
◆
�
�
�
�
C
8
<
:
X
|�|2
||@�g||�
9
=
;
{||Ef � Eg||+ ||Bf �Bg||}||f � g||�
C
8
<
:
X
|�|2
||@�g||2�
9
=
;
{||Ef � Eg||2 + ||Bf �Bg||2}+1
4||f � g||2�
C|||@�g|||2{||Ef � Eg||2 + ||Bf �Bg||2}+1
4||f � g||2�
CM0
{||Ef � Eg||2 + ||Bf �Bg||2}+1
4||f � g||2�.
Similarly, we use the Sobolev embedding theorem as well as elementary inequalities
to estimate the terms below
�
�
�
�
✓
Ef ·p
p0
(f � g), f � g
◆
�
�
�
�
Cp
M0
||f � g||2
�
�
�
�
✓
u{Ef � Eg} ·p
p0
g, f � g
◆
�
�
�
�
C||f � g||�||Ef � Eg||X
|�|2
||@�g||�
1
4||f � g||2� + CM
0
||Ef � Eg||2.
By Theorem 2.4 as well as (2.14),
|(�(f � g, f) + �(g, f � g), f � g)|
C {kf � gkkfk� + kf � gk�kfk+ kf � gkkgk� + kf � gk�kgk} kf � gk�
= C {kfk+ kgk} kf � gk2
� + C {kfk� + kgk�} kf � gkkf � gk�
Cp
M0
kf � gk2
� + C�
kfk2
� + kgk2
�
kf � gk2 +1
4kf � gk2
�
Cp
M0
kf � gk2
� + CM0
kf � gk2 +1
4kf � gk2
�
From the Maxwell system in (2.93), we deduce from (2.88) that
�⇣
2(Ef � Eg) · (pp
J/p0
)⇠1
, f � g⌘
=d
dt{||Ef � Eg||2 + ||Bf �Bg||2}.
63
By taking the inner product of (2.93) with f � g, and collecting above estimates as
well as plugging in the K and A estimates from Lemma 2.8, we have
d
dt
⇢
1
2||f � g||2 + ||Ef � Eg||2 + ||Bf �Bg||2
�
+ ||f � g||2�
C{M0
+p
M0
+ 1}{||f � g||2 + ||Ef � Eg||2 + ||Bf �Bg||2}
+
✓
C
m+ C
p
M0
+3
4
◆
||f � g||2�.
If we choose m and M0
so that Cm
+ Cp
M0
< 1
4
then the last term on the r.h.s. can
be absorbed by ||f � g||2� from the right. We deduce f(t) ⌘ g(t) from the Gronwall
inequality.
To show the continuity of E(f(t)) with respect to t, we have from (2.92) that as
t ! s
|E(t)� E(s)| CM0
(t� s) + C
✓
sups⌧t
E1/2(⌧) + 1
◆
Z t
s
|||f |||2�(⌧)d⌧ ! 0.
For the positivity of F = J + J1/2f , since fn solves (2.84), we see that F n =
J + J1/2fn solves the iterating sequence (n � 0):
⇢
@t +p
p0
·rx + ⇠
✓
En +p
p0
⇥Bn
◆
·rp
�
F n+1 = Cmod(F n+1, F n)
together with the coupled Maxwell system:
@tEn �rx ⇥Bn = �J n = �
Z
R3
p
p0
{F n+
� F n�}dp,
@tBn +rx ⇥ En = 0, rx ·Bn = 0,
rx · En = ⇢n =
Z
R3
{F n+
� F n�}dp.
And, as in (2.84), the first step in the iteration is given through the initial data
F 0(t, x, p) = [F 0
+
(t, x, p), F 0
�(t, x, p)] = [F0,+(x, p), F
0,�(x, p)]
= [J + J1/2f0,+(x, p), J + J1/2f
0,�(x, p)].
64
Above we have used the modification Cmod = [Cmod+
, Cmod� ] where
Cmod± (F n+1, F n) = @p
i
@pj
F n+1
± (p)
Z
R3
�ij(P, Q)�
F n+
+ F n��
dq
+@pj
F n+1
± (p)
Z
R3
@pi
�ij(P, Q)�
F n+
+ F n��
dq
�@pi
F n+1
± (p)
Z
R3
�ij(P, Q)@qj
�
F n+
+ F n��
dq
�F n±(p)@p
i
Z
R3
�ij(P, Q)@qj
�
F n+
+ F n��
dq.
Since F 0(t, x, p) � 0 Lemma 2.4, the elliptic structure of this iteration and the
maximum principle imply that F n+1(t, x, p) � 0 if F n(t, x, p) � 0. This implies
F (t, x, p) � 0.
Finally, since E(t) < +1, [f, @2E, @2B] is bounded and continuous. By F =
J +J1/2f, it is straightforward to verify that classical mass, total mometum and total
energy conservations hold for such solutions constructed. We thus conclude Theorem
2.6.
2.5. Positivity of the Linearized Landau Operator
We establish the positivity of the linear operator L for any small amplitude solu-
tion [f(t, x, p), E(t, x), B(t, x)] to the full relativistic Landau-Maxwell system (2.29)
and (2.30). Recall the orthogonal projection Pf with coe�cients a±, b and c in (2.19).
For solutions to the nonlinear system, Lemmas 2.11 and 2.12 are devoted to basic
estimates for the linear and nonlinear parts in the macroscopic equations. We make
the crucial observation in Lemma 2.13 that the electromagnetic field roughly speak-
ing is bounded by ||f ||�(t) at any moment t. Then based on Lemma 2.10, we finally
establish Theorem 2.2 by a careful study of macroscopic equations coupled with the
Maxwell system.
We begin with a formal definition of the orthogonal projection P. Define
⇢0
=
Z
R3
J(p)dp, ⇢i =
Z
R3
p2
i J(p)dp (i = 1, 2, 3),
⇢4
=
Z
R3
|p|2J(p)dp, ⇢5
=
Z
R3
p0
J(p)dp.
65
We can write an orthonormal basis for N in (2.15) with normalized constants as
✏⇤1
= ⇢�1/2
0
[J1/2, 0], ✏⇤2
= ⇢�1/2
0
[0, J1/2],
✏⇤i+2
= (2⇢i)�1/2[piJ
1/2, piJ1/2] (i = 1, 2, 3), ✏⇤
6
= c6
✓
[p0
, p0
]� ⇢5
⇢0
[1, 1]
◆
J1/2
where c�2
6
= 2(⇢0
+⇢4
)�2⇢25
⇢0. Now consider Pf , f = [f
+
, f�], we define the coe�cients
in (2.19) so that P is an orthogonal projection:
a+
⌘ ⇢�1/2
0
hf, ✏⇤1
i � ⇢5
⇢0c, a� ⌘ ⇢�1/2
0
hf, ✏⇤2
i � ⇢5
⇢0c,
bj ⌘ (2⇢j)�1/2hf, ✏⇤j+2
i, c ⌘ c6
hf, ✏⇤6
i.(2.94)
Proposition 2.2. Let @� = @�0t @
�1x1@�2
x2@�3
x3. There exists C > 1 such that
1
C||@�Pf ||2� ||@�a±||2 + ||@�b||2 + ||@�c||2 C||@�Pf ||2.
For the rest of the section, we concentrate on a solution [f, E, B] to the nonlinear
relativistic Landau-Maxwell system.
Lemma 2.10. Let [f(t, x, p), E(t, x), B(t, x)] be the solution constructed in Theo-
rem 2.6 to (2.29) and (2.30), which satisfies (2.31), (2.10), (2.11) and (2.12). Then
we have
2
3⇢
4
Z
T3
b(t, x) =
Z
T3
B(t, x)⇥ E(t, x),(2.95)
�
�
�
�
Z
T3
a+
(t, x)
�
�
�
�
+
�
�
�
�
Z
T3
a�(t, x)
�
�
�
�
+
�
�
�
�
Z
T3
c(t, x)
�
�
�
�
C�
||E||2 + ||B � B||2�
,(2.96)
where a = [a+
, a�], b = [b1
, b2
, b3
], c are defined in (2.94).
Proof. We use the conservation of mass, momentum and energy. For fixed (t, x),
notice that (2.94) implies
Z
p{f+
+ f�}p
Jdp =2
3b(t, x)
Z
|p|2Jdp.
Hence (2.95) follows from momentum conservation (2.11) with normalized constants.
66
On the other hand, for fixed (t, x), (2.94) impliesZ
f±p
Jdp = ⇢0
a±(t, x) + ⇢5
c(t, x),
Z
p0
{f+
+ f�}p
Jdp = ⇢5
{a+
(t, x) + a�(t, x)}+ 2(⇢0
+ ⇢4
)c(t, x),
Upon further integration over T3, we deduce from the mass conservation (2.10) thatR
T3 a+
=R
T3 a� = �⇢5
⇢0
R
T3 c. From the reduced energy conservation (2.12),
�Z
T3
{|E(t)|2 + |B(t)� B|2} = 2
✓
⇢0
+ ⇢4
� ⇢2
5
⇢0
◆
Z
T3
c.
By the sharp form of Holder’s inequality, (⇢0
+ ⇢4
)⇢0
� ⇢2
5
> 0.
⇤
We now derive the macroscopic equations for Pf ’s coe�cients a±, b and c. Recall-
ing equation (2.16) with (2.17) and (2.18) with normalized constants in (2.104) and
(2.105), we further use (2.19) to expand entries of l.h.s. of (2.16) as⇢
@0a± +pj
p0
�
@ja± ⌥ Ej
+pjpi
p0
@ibj + pj
�
@0bj + @jc
+ p0
@0c
�
J1/2(p)
where @0 = @t and @j = @xj
. For fixed (t, x), this is an expansion of l.h.s. of (2.16)
with respect to the basis of (1 i, j 3)
[p
J, 0], [0,p
J ], [pj
pJ/p
0
, 0], [0, pj
pJ/p
0
],
[pj
pJ, pj
pJ ], [pjpi
pJ/p
0
, pjpi
pJ/p
0
], [p0
pJ, p
0
pJ ](2.97)
Expanding the r.h.s. of (2.16) with respect to the same basis (2.97) and comparing
coe�cients on both sides, we obtain the important macroscopic equations for a(t, x) =
[a+
(t, x), a�(t, x)], bi(t, x) and c(t, x):
@0c = lc + hc,(2.98)
@ic + @0bi = li + hi,(2.99)
(1� �ij)@ibj + @jbi = lij + hij,(2.100)
@ia± ⌥ Ei = l±ai + h±ai,(2.101)
@0a± = l±a + h±a .(2.102)
67
Here lc(t, x), li(t, x), lij(t, x), l±ai(t, x) and l±a (t, x) are the corresponding coe�cients
of such an expansion of the linear term l({I�P}f), and hc(t, x), hi(t, x), hij(t, x),
h±ai(t, x) and h±a (t, x) are the corresponding coe�cients of the same expansion of the
higher order term h(f).
From (2.19) and (2.94) we see thatZ
[pp
J/p0
,�pp
J/p0
] ·Pfdp = 0,
Z
[p
J,�p
J ] · fdp = ⇢0
{a+
� a�}.
We plug this into the coupled maxwell system, (2.30) and (2.31), to obtain
@tE �rx ⇥B = �J =
Z
R3
[pp
J/p0
,�pp
J/p0
] · {I�P}fdp,(2.103)
@tB +rx ⇥ E = 0, rx · E = ⇢0
{a+
� a�}, rx ·B = 0.
We rewrite the terms (2.17) and (2.18) in (2.16) with normalized constants as
l({I�P}f) ⌘ �⇢
@t +p
p0
·rx + L
�
{I�P}f,(2.104)
h(f) ⌘ �⇠✓
E +p
p0
⇥B
◆
·rpf +⇠
2
⇢
E · p
p0
�
f + �(f, f).(2.105)
Next, we estimate these terms.
Lemma 2.11. For any 1 i, j 3,
X
|�|N�1
||@�lc||+ ||@�li||+ ||@�lij||+ ||@�l±ai||+ ||@�l±a ||+ ||@�J ||
CX
|�|N
k{I�P}@�fk.
Proof. Let {✏n(p)} represent the basis in (2.97). For fixed (t, x), we can use the
Gram-Schmidt procedure to argue that the terms lc(t, x), li(t, x), lij(t, x), l±ai(t, x) and
l±a (t, x) are of the form18
X
n=1
cnhl({I�P}f), ✏ni,
where cn are constants which do not depend on on f . Let |�| N � 1. By (2.104)Z
@�l({I�P}f) · ✏n(p)dp = �Z
⇢
@t +p
p0
·rx + L
�
{I�P}@�f(p) · ✏n(p)dp.
68
We estimate the first two terms,
kZ
{@t +p
p0
·rx}({I�P}@�f) · ✏ndpk2
2
Z
|✏n|dp⇥Z
T3⇥R3
|✏n(p)|(|{I�P}@0@�f |2 + |{I�P}rx@�f |2)dpdx
C�
||{I�P}@0@�f ||2 + ||{I�P}rx@�f ||2
�
.
Similarly, we have
||@�J || = ||Z
R3
[�pp
J/p0
, pp
J/p0
] · {I�P}@�fdp|| C||{I�P}@�f ||.
Using (2.72) we can estimate the last term
khL{I�P}@�f, ✏nik Ck{I�P}@�fk.
Indeed (2.72) was designed to estimate this term. ⇤
We now estimate coe�cients of the higher order term h(f).
Lemma 2.12. Let (2.14) be valid for some M0
> 0. Then
X
|�|N
{||@�hc||+ ||@�hi||+ ||@�hij||+ ||@�h±ai||+ ||@�h±a ||} Cp
M0
X
|�|N
||@�f ||�.
Proof. Let |�| N , recall that {✏n(p)} represents the basis in (2.97). Notice
that @�hc, @�hi, @�hij, @�h±ai and @�h±a are again of the form
18
X
n=1
cnh@�h(f), ✏ni.
It again su�ces to estimate h@�h(f), ✏ni. For the first term of h(f) in (2.105), we use
an integration by parts over the p variables to get
�Z
@�{⇠(E +p
p0
⇥B) ·rpf)} · ✏n(p)dp
= �X
C�1�
Z
rp · {⇠(@�1E +p
p0
⇥ @�1B)@���1f} · ✏n(p)dp
=X
C�1�
Z
⇠(@�1E +p
p0
⇥ @�1B)@���1f ·rp✏n(p)dp
CX
{|@�1E|+ |@�1B|}⇢
Z
|@���1f |2dp
�
1/2
.
69
The last estimate holds because rp✏n(p) has exponential decay. Take the square of
the above, whose further integration over T3 is bounded by
(2.106) C
Z
T3
{|@�1E|+ |@�1B|}2
⇢
Z
|@���1f |2dp
�
dx.
If |�1
| N/2, by H2(T3) ⇢ L1(T3) and the small amplitude assumption (2.14), we
have
supx{|@�1E|+ |@�1B|} C
X
|�|N
{||@�E(t)||+ ||@�B(t)||} Cp
M0
.
If |�1
| � N/2 thenR
T3{|@�1E|+ |@�1B|}2dx M0
and, by (2.90),
supx
⇢
Z
|@���1f |2dp
�
CX
|�|N
||@�f(t)||2.
We thus conclude that (2.106) is bounded by Cp
M0
P
|�|N ||@�f ||.
The second term of h(f) in (2.105) is easily treated by the same argument, forZ
⇠
2@�{(E · p
p0
)f} · ✏n(p)dp
=X
C�1�
Z
{⇠2(@�1E · p
p0
)@���1f} · ✏n(p)dp
CX
|@�1E|⇢
Z
|@���1f |2dp
�
1/2
.
For the third term of h(f) in (2.105) we apply (2.71):
kh@��(f, f), ✏nik CX
|�|N
||@�f(t)||X
|�|N
||@�f(t)||� Cp
M0
X
|�|N
||@�f ||�.
We designed (2.71) to estimate this term. ⇤
Next we estimate the electromagnetic field [E(t, x), B(t, x)] in terms of f(t, x, p)
through the macroscopic equation (2.101) and the Maxwell system (2.103).
Lemma 2.13. Let [f(t, x, p), E(t, x), B(t, x)] be the solution to (2.29), (2.30) and
(2.31) constructed in Theorem 2.6. Let the small amplitude assumption (2.14) be
valid for some M0
> 0. Then there is a constant C > 0 such that
X
|�|N�1
{||@�E(t)||+ ||@�{B(t)� B}||} CX
|�|N
⇣
||@�f(t)||+p
M0
||@�f(t)||�⌘
.
70
Proof. We first use the plus part of the macroscopic equation (2.101) to estimate
the electric field E(t, x) :
�@�Ei = @�l+ai + @�h+
ai � @�@ia+
.
Proposition 2.2 says ||@�@ia+
|| C||P@�@if ||. Applying Lemmas’ 2.11 and 2.12 to
@�l+ai and @�h+
ai respectively, we deduce that for |�| N � 1,
(2.107) ||@�E|| CX
|�0|N
⇣
||@�0f(t)||+p
M0
||@�0f(t)||�⌘
.
We next estimate the magnetic field B(t, x). Let |�| N � 2. Taking @� to the
Maxwell system (2.103) we obtain
rx ⇥ @�B = @�J + @t@�E, rx · @�B = 0.
Lemma 2.11, (2.107) as well asR
|r ⇥ @�B|2 + (r · @�B)2dx =R
P
i,j(@xi
@�Bj)2dx
imply
||r@�B|| C{||@�J ||+ ||@t@�E||} C
X
|�0|N
⇣
||@�0f(t)||+p
M0
||@�0f(t)||�⌘
By @t@�B +r⇥@�E = 0, ||@t@�B|| ||r⇥@�E||. Finally, by the Poincare inequality
||B � B|| C||rB||, we therefore conclude our Lemma. ⇤
We now prove the crucial positivity of L for a small solution [f(t, x, p), E(t, x), B(t, x)]
to the relativistic Landau-Maxwell system. The conservation laws (2.10), (2.11) and
(2.12) play an important role.
Proof of Theorem 2.2. From (2.77) we have
(L@�f, @�f) � �||{I�P}@�f ||2�.
By Proposition 2.2, we need only establish (2.20). The rest of the proof is devoted to
establishing
X
|�|N
{||@�a±||+ ||@�b||+ ||@�c||}
CX
|�|N
||{I�P}@�f(t)||+ Cp
M0
X
|�|N
||@�f(t)||�,(2.108)
71
This is su�cient to prove the upper bound in (2.20) because the second term on the
r.h.s. can be neglected for M0
small:
X
|�|N
||@�f(t)||� X
|�|N
||P@�f(t)||� +X
|�|N
||{I�P}@�f(t)||�
CX
|�|N
(||@�a±||+ ||@�b||+ ||@�c||) +X
|�|N
||{I�P}@�f(t)||�.
We will estimate each of the terms a±, b and c in (2.108) one at a time.
We first estimate r@�b. Let |�| N � 1. From (2.100)
�@�bj + @j(r · @�b) =X
i
@i�
@�@ibj + @�@jbi
�
=X
i
@i@� (lij + hij) (1 + �ij)
Multiplying with @�bj and summing over j yields:
Z
T3
(
(r · @�b)2 +X
i,j
�
@i@�bj
�
2
)
dx
=X
i,j
Z
T3
(@�lij + @�hij) (1 + �ij)@i@�bjdx.
Therefore
X
i,j
||@i@�bj||2 C
X
i,j
||@i@�bj||!
X
{||@�lij||+ ||@�hij||},
which implies, using⇣
P
i,j ||@i@�bj||⌘
2
CP
i,j ||@i@�bj||2, that
(2.109)X
i,j
||@i@�bj|| CX
{||@�lij||+ ||@�hij||}.
This is bounded by the r.h.s. of (2.108) by Lemmas 2.11 and 2.12. We estimate
purely temporal derivatives of bi(t, x) with � = [�0, 0, 0, 0] and 0 < �0 N � 1. From
(2.98) and (2.99), we have
@0@�bi = @�li + @�hi � @i@�c
= @�li + @�hi � @�0@0c
= @�li + @�hi � @�0lc � @�0hc,
72
where |�0| = �0. Therefore,
k@0@�bik C⇣
k@�lik+ k@�hik+ k@�0lck+ k@�0hck⌘
.
By Lemmas 2.11 and 2.12, this is bounded by the r.h.s. of (2.108). Next, assume
0 �0 1. We use the Poincare inequality and (2.95) to obtain
||@�0
t bi|| C
⇢
||r@�0
t bi||+�
�
�
�
@�0
t
Z
bi(t, x)dx
�
�
�
�
�
= C
⇢
||r@�0
t bi||+�
�
�
�
@�0
t
Z
E ⇥Bdx
�
�
�
�
�
.
By (2.109), it su�ces to estimate the last term above. From Lemma 2.13 and the
assumption (2.14), with M0
1, the last term is bounded by
||@�0
t B|| · ||E||+ ||B|| · ||@�0
t E||
p
M0
CX
|�|N
⇣
||@�f(t)||+p
M0
||@�f(t)||�⌘
Cp
M0
X
|�|N
||@�f(t)||�.
We thus conclude the case for b.
Now for c(t, x), from (2.98) and (2.99),
||@0@�c|| C{||@�lc||+ ||@�hc||},
||r@�c|| C||@0@�bi||+ ||@�li||+ ||@�hi||.
Thus, for |�| N�1, both ||@0@�c|| and ||r@�c|| are bounded by the r.h.s. of (2.108)
by the above argument for b and Lemmas 2.11 and 2.12. Next, to estimate c(t, x)
itself, from the Poincare inequality and Lemma 2.10
||c|| C
⇢
||rc||+�
�
�
�
Z
cdx
�
�
�
�
�
C{||rc||+ ||E||2 + ||B � B||2}.73
Notice that from (2.3) and Jensen’s inequality |B| kBk. Using this, Lemma 2.13
and (2.14), with M0
1, imply
||E||2 + ||B � B||2 ||E||2 + C||B � B||(||B||+ ||B||) Cp
M0
X
|�|N
||@�f(t)||�.
We thus complete the estimate for c(t, x) in (2.108).
Now we consider a(t, x) = [a+
(t, x), a�(t, x)]. By (2.102),
||@t@�a±|| C
�
||@�l±a ||+ ||@�h±a ||
.
We now use Lemma 2.11 and 2.12, for |�| N � 1, to say that ||@t@�a|| is bounded
by the r.h.s. of (2.108). We now turn to purely spatial derivatives of a(t, x). Let
|�| N � 1 and � = [0, �1
, �2
, �3
] 6= 0. By taking @i of (2.101) and summing over i
we get
(2.110) ��@�a± ±r · @�E = �X
i
@i@�{l±ai + h±ai}.
But from the Maxwell system in (2.103),
r · @�E = ⇢0
(@�a+
� @�a�).
Multiply (2.110) with @�a± so that the ± terms are the same and integrate over T3.
By adding the ± terms together we have
||r@�a+
||2 + ||r@�a�||2 + ⇢0
||@�a+
� @�a�||2
C{||r@�a+
||+ ||r@�a�||}X
±||@�{l±bi + h±bi}||.
Therefore, ||r@�a+
||+ ||r@�a�|| P
± ||@�{l±bi +h±bi}||. Since � is purely spatial, this
is bounded by the r.h.s of (2.108) because of Lemmas 2.11 and 2.12. Furthermore,
by the Poincare inequality and Lemma 2.10, a itself is bounded by
||a±|| C||ra±||+ C
�
�
�
�
Z
a±dx
�
�
�
�
C||ra±||+ C{||E||2 + ||B � B||2},
which is bounded by the r.h.s. of (2.108) by the same argument as for c. We thus
complete the estimate for a(t, x) and our theorem follows.
74
2.6. Global Solutions
In this section we establish Theorem 2.1. We first derive a refined energy estimate
for the relativistic Landau-Maxwell system.
Lemma 2.14. Let [f(t, x, p), E(t, x), B(t, x)] be the unique solution constructed in
Theorem 2.6 which also satisfies the conservation laws (2.10), (2.11) and (2.12). Let
the small amplitude assumption (2.14) be valid. For any given 0 m N, |�| m,
there are constants C|�| > 0, C⇤m > 0 and �m > 0 such that
X
|�|m,|�|+|�|N
1
2
d
dtC|�|||@�
�f(t)||2 +1
2
d
dt|||[E, B]|||2(t)
+X
|�|m,|�|+|�|N
�m||@��f(t)||2� C⇤
m
p
E(t)|||f |||2�(t).(2.111)
Proof. We use an induction over m, the order of the p�derivatives. For m = 0,
by taking the pure @� derivatives of (2.29), we obtain:
⇢
@t +p
p0
·rx + ⇠
✓
E +p
p0
⇥B
◆
·rp
�
@�f
�⇢
@�E · p
p0
�pJ⇠
1
+ L{@�f}(2.112)
= �X
�1 6=0
C�1� ⇠
✓
@�1E +p
p0
⇥ @�1B
◆
·rp@���1f
+X
�1�
C�1�
⇢
⇠
2
⇢
@�1E · p
p0
�
@���1f + �(@�1f,@���1f)
�
Using the same argument as (2.88),
�h@�E · {pp
J/p0
}⇠1
, @�fi =1
2
d
dt
�
||@�E(t)||2 + ||@�B(t)||2
.
Take the inner product of @�f with (2.112), sum over |�| N and apply Theorem
2.2 to L{@�f} to deduce the following for some constant C > 0,
X
|�|N
1
2
d
dt
�
||@�f(t)||2 + ||@�E(t)||2 + ||@�B(t)||2�
+ �0
X
|�|N
||@�f(t)||2�
C{|||f |||(t) + |||[E, B]|||(t)}|||f |||2�(t) Cp
E(t)|||f |||2�(t).
75
We have used estimates (2.89-2.91) and Theorem 2.4 to bound the r.h.s. of (2.112).
This concludes the case for m = 0 with C0
= 1 and C⇤0
= C.
Now assume the Lemma is valid for m. For |�| = m + 1, taking @��(� 6= 0) of
(2.29), we obtain:
⇢
@t +p
p0
·rx + ⇠
✓
E +p
p0
⇥B
◆
·rp
�
@��f � @�E · @�
⇢
p
p0
pJ
�
⇠1
+@�{L@�f}+X
�1 6=0
C�1� @�1
✓
p
p0
◆
·rx@����1
f(2.113)
=X
C�1� C�1
�
⇠
2
⇢
@�1E · @�1
✓
p
p0
◆�
@���1���1
f �X
�1 6=0
C�1� ⇠@
�1E ·rp@���1� f
�X
(�1,�1) 6=(0,0)
C�1� C�1
� ⇠@�1
✓
p
p0
◆
⇥ @�1B ·rp@���1���1
f
+X
C�1� @��(@�1f,@
���1f).
We take the inner product of (2.113) over T3⇥R3 with @��f . The first inner product
on the left is equal to 1
2
ddt||@�
�f(t)||2. Now |�| N�1 (since |�| = m+1 > 0), Lemma
2.13, (2.60) and M0
1 imply (after an integration by parts) that the second inner
product on l.h.s. is bounded by
h@�E · @�{pp
J/p0
}⇠1
, @��fi C||@�E|| · ||@�f || C||@�f ||
X
|�0|N
||@�0f ||�.
From Lemma 2.7 and Cauchy’s inequality we deduce that, for any ⌘ > 0, the inner
product of third term on l.h.s. is bounded from below as
�
@�{L@�f}, @��f�
� ||@��f ||2� � ⌘
X
|¯�||�|||@�
¯�f ||2� � C⌘||@�f ||2.
Using Cauchy’s inequality again, the the inner product of the last term on l.h.s. of
(2.113) is bounded by
⌘||@��f(t)||2 + C⌘
X
|�1|�1
||rx@����1
f ||2.
By the same estimates, (2.89-2.91) and Theorem 2.4, all the inner products in r.h.s.
of (2.113) are bounded by Cp
E(t)|||f |||2�(t). Collecting terms and summing over
76
|�| = m+1 and |�|+ |�| N , we split the highest order p-derivatives from the lower
order derivatives to obtain
X
|�|=m+1,|�|+|�|N
⇢
1
2
d
dt||@�
�f(t)||2 + ||@��f(t)||2�
�
X
|�|=m+1,|�|+|�|N
8
<
:
X
|�|=m+1
2⌘||@��f(t)||2� + C
p
E(t)|||f |||2�(t)
9
=
;
+X
|�|=m+1,|�|+|�|N
(C + 2C⌘)X
|�|m,|�|+|�|N
||@��f(t)||2�
Zm+1
8
<
:
X
|�|=m+1,|�|+|�|N
2⌘||@��f(t)||2� + C
p
E(t)|||f |||2�(t)
9
=
;
+Zm+1
(C + 2C⌘)X
|�|m,|�|+|�|N
||@��f(t)||2�.
Here Zm+1
denotes the number of all possible (�, �) such that |�| m+1, |�|+|�| N
. By choosing ⌘ = 1
4Zm+1
, and absorbing the first term on the r.h.s. by the second
term on the left, we have, for some constant C(Zm+1
),
X
|�|=m+1,|�|+|�|N
⇢
1
2
d
dt||@�
�f(t)||2 +1
2||@�
�f(t)||2��
C(Zm+1
)
8
<
:
X
|�|m,|�|+|�|N
||@��f(t)||2� +
p
E(t)|||f |||2�(t)
9
=
;
.(2.114)
We may assume C(Zm+1
) � 1. We multiply (2.114) by �m
2C(Zm+1)
and add it to (2.111)
for |�| m to get
X
|�|=m+1,|�|+|�|N
⇢
�m4C(Zm+1
)
d
dt||@�
�f(t)||2 +�m
4C(Zm+1
)||@�
�f(t)||2��
+X
|�|m,|�|+|�|N
1
2
d
dt
�
C|�|||@��f(t)||2 + ||@�E(t)||2 + ||@�B(t)||2
�
+X
|�|m,|�|+|�|N
�m||@��f(t)||2�
�m2
X
|�|m,|�|+|�|N
||@��f(t)||2� +
⇢
C⇤m +
�m2
�
p
E(t)|||f |||2�(t).
77
Absorb the first term on the right by the last term on the left. We conclude our
lemma by choosing
Cm+1
=�m
4C(Zm+1
), �m+1
=�m
4C(Zm+1
) �m
2, C⇤
m+1
= C⇤m +
�m2
.
Nothing that C(Zm+1
) > C(Zm) and �m < �m�1
. ⇤
We are ready to construct global in time solutions to the relativistic Landau-
Maxwell system (2.29) and (2.30).
Proof of Theorem 2.1: We first fix M0
1 such that both Theorems 2.2 and 2.6 are
valid. For such an M0
, we let m = N in (2.111), and define
y(t) ⌘X
|�|+|�|N
C|�|||@��f(t)||2 + |||[E, B]|||2(t).
We choose a constant C1
> 1 such that for any t � 0,
1
C1
⇢
y(t) +�N2
Z t
0
|||f |||2�(s)ds
�
E(t)
E(t) C1
⇢
y(t) +�N2
Z t
0
|||f |||2�(s)ds
�
.
Recall constant C⇤N in (2.111). We define
M ⌘ min
⇢
�2
N
8C⇤2N C2
1
,M
0
2C2
1
�
,
and choose initial data so that E(0) M < M0
. From Theorem 2.6, we may denote
T > 0 so that
T = supt{t : E(t) 2C2
1
M} > 0.
Notice that, for 0 t T, E(t) 2C2
1
M M0
so that the small amplitude assump-
tion (2.14) is valid. We now apply Lemma 2.14 and the definitions of M and T , with
0 t T , to get
y0(t) + �N |||f |||2�(t)
C⇤N
p
E(t)|||f |||2�(t) C⇤NC
1
p2M |||f |||2�(t)
�N2|||f |||2�(t).
78
Therefore, an integration in t over 0 t s < T yields
E(s) C1
⇢
y(s) +�N2
Z s
0
|||f |||2�(⌧)d⌧
�
C1
y(0)
C2
1
E(0)(2.115)
C2
1
M < 2C2
1
M.
Since E(s) is continuous in s, this implies E(T ) C2
1
M if T < 1. This implies
T = 1. Furthermore, such a global solution satisfies E(t) C2
1
E(0) for all t � 0 from
(2.115). Q.E.D.
79
CHAPTER 3
Almost Exponential Decay near Maxwellian
Abstract. By direct interpolation of a family of smooth energy estimates for
solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann
type equations in [37–39,62], we show convergence to Maxwellian with any polynomial
rate in time. Our results not only resolve the important open problem for both the
Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for
charged particles, but also lead to a simpler alternative proof of recent decay results
[21] for soft potentials as well as the Coulombic interaction, with precise decay rate
depending on the initial conditions. This result will appear in a modified form as
[60].
3.1. Introduction
There are two motivations of the current study. Recently, a new nonlinear energy
method has been developed to construct global-in-time solutions near Maxwellian for
various types of Boltzmann equations. One of the highlights of such a program is the
construction of global-in-time solutions for the Boltzmann equation in the presence
of a self-consistent electromagnetic field. Unfortunately, the question of determining
a possible time-decay rate for the electromagnetic field has been left open. For near
Maxwellian periodic solutions to the Boltzmann equation with no electromagnetic
field, Ukai [65] obtained exponential convergence in the case of a cuto↵ hard potential.
Caflisch [8,9], for cuto↵ soft potentials with � > �1, obtained a convergence rate like
O(e��t�) for � > 0 and 0 < � < 1. In the whole space, also for cuto↵ soft potentials
with � > �1, Ukai and Asano [66] obtained the rate O(t�↵) with 0 < ↵ < 1. However
the decay for very soft potentials had been open.
80
Desvillettes and Villani have recently undertaken an impressive program to study
the time-decay rate to a Maxwellian of large data solutions to Boltzmann type equa-
tions. Even though their assumptions in general are a priori and impossible to verify
at present time, their method does lead to an almost exponential decay rate for the
soft potentials and the Landau equation for solutions close to a Maxwellian. This
surprising new decay result relies crucially on recent energy estimates in [37, 38] as
well as other extensive and delicate work [18–21, 53, 64, 69]. To obtain time decay
for these models with very ‘weak’ collision e↵ects has been a very challenging open
problem even for solutions near a Maxwellian, therefore it is of great interest to find
simpler proofs for such decay.
The objective of this article is to give a direct proof of an almost exponential
decay rate for solutions near Maxwellian to the Vlasov-Maxwell-Boltzmann system,
the relativistic Landau-Maxwell system, the Boltzmann equation with a soft potential
as well as the Landau equation. The common di�culty of all such problems is the
fact that in the energy estimates [37–39, 62] the instant energy functional at each
time is stronger than the dissipation rate. It is thus impossible to use a Gronwall
type of argument to get the decay rate in time. Our main observation is that a family
of energy estimates, not just one, have been derived in [37–39, 62]. Even though
the instant energy is stronger for a fixed family, it is possible to be bounded by a
fractional power of the dissipation rate via simple interpolations with stronger energy
norms of another family of estimates. Only algebraic decay is possible due to such
interpolations, but more regular initial data grants faster decay. In comparison to
[21], our proofs are much simpler and our decay rates are more precise.
For both Vlasov-Maxwell-Boltzmann and relativistic Landau-Maxwell, such a
family of energy norms are exactly the same as in [39, 62] with higher and higher
derivatives. The interpolation in this case is between Sobolev norms.
On the other hand, for both soft potentials and Landau equations, such a family
of energy norms depends on higher and higher powers of velocity weights and the
interpolation in this case is between the di↵erent weight powers. Although these are
81
di↵erent from the original norms used in [37,38], new energy estimates can be derived
by the improved method in [39,62] with minimum modifications.
We remark that all the constants in general are not explicit, except for the poly-
nomial decay exponent. This is due to the fact that the lower bound for the linearized
collision operator is not obtained constructively.
3.2. Vlasov-Maxwell-Boltzmann
Dynamics of charged dilute particles (e.g., electrons and ions) are described by
the Vlasov-Maxwell-Boltzmann system:
@tF+
+ v ·rxF+
+ (E + v ⇥B) ·rvF+
= Q(F+
, F+
) + Q(F+
, F�),
(3.1)
@tF� + v ·rxF� � (E + v ⇥B) ·rvF� = Q(F�, F+
) + Q(F�, F�),
with initial data F±(0, x, v) = F0,±(x, v). For notational simplicity we have set all
the physical constants to be unity, see [39] for more background. Here F±(t, x, v) � 0
are the spatially periodic number density functions for the ions (+) and electrons (-)
respectively at time t � 0, position x = (x1
, x2
, x3
) 2 [�⇡, ⇡]3 = T3 and velocity
v = (v1
, v2
, v3
) 2 R3. As a model problem, the collision between particles is given by
the standard Boltzmann collision operator with hard-sphere interaction Q in [39].
The self-consistent, spatially periodic electromagnetic field [E(t, x), B(t, x)] in
(3.1) is coupled with F±(t, x, v) through the Maxwell system:
(3.2) @tE �rx ⇥B = �J , @tB +rx ⇥ E = 0,
with constraints rx · B = 0, rx · E = ⇢ and initial data E(0, x) = E0
(x), B(0, x) =
B0
(x). The coupling comes through the charge density ⇢ and the current density J
which are given by
(3.3) ⇢ =
Z
R3
{F+
� F�}dv, J =
Z
R3
v{F+
� F�}dv.
We consider the perturbation F± = µ +p
µf± � 0 where the normalized Maxwellian
is µ = µ(v) = e�|v|2.
82
Notation: Let || · || be the standard norm in either L2(T3 ⇥ R3) or L2(T3). We
also define || · ||⌫ to be the following weighted L2 norm
||g||2⌫ ⌘Z
T3⇥R3
⌫(v)|g(x, v)|2dxdv,
where ⌫(v) is the collision frequency for hard-sphere interactions, ⌫(v) ⇠ C|v| as |v|
tends to 1. Let the multi-indices ↵ and � be ↵ = [↵0,↵1,↵2,↵3], � = [�1, �2, �3]
with length |↵| and |�| respectively. Then let @↵� ⌘ @↵0
t @↵1
x1@↵2
x2@↵3
x3@�1
v1@�2
v2@�3
v3, which
includes temporal derivatives. Define
|||f |||2N(t) ⌘X
|↵|+|�|N
||@↵� f
+
(t)||2 +X
|↵|+|�|N
||@↵� f�(t)||2,(3.4)
|||[E, B]|||2N(t) ⌘X
|↵|N
||@↵E(t)||2 +X
|↵|N
||@↵B(t)||2.
Given initial datum [f0
(x, p), E0
(x), B0
(x)], we define
(3.5) EN(0) =1
2|||f
0
|||2N + |||[E0
, B0
]|||2N ,
where the temporal derivatives of [f0
, E0
, B0
] are defined naturally through equations
(3.1) and (3.2). Further define the the dissipation rate as
|||f |||2N,⌫(t) ⌘X
|↵|+|�|N
||@↵� f
+
||2⌫ +X
|↵|+|�|N
||@↵� f�||2⌫ .
We study the decay of solutions near F± ⌘ µ, E ⌘ 0, and B ⌘ B = a constant.
Theorem 3.1. Fix integers N � 4 and k � 1. Choose initial data [f0,±, E
0
, B0
]
which satisfies the assumptions of Theorem 1 in [39] for N + k (therefore also for N)
including for ✏N+k > 0 small enough
EN+k(0) ✏N+k,
and let [F± = µ +p
µf±, E, B] be the unique solution to (3.1) and (3.2) with (3.3)
from Theorem 1 of [39]. Then there exists CN,k > 0 such that
|||f |||2N(t) + |||[E, B � B]|||2N(t) CN,kEN+k(0)
⇢
1 +t
k
��k
,
where B = 1
|T3|R
T3 B(t, x)dx is constant in time [39].
83
Proof. On the last page of [39], we see the following inequality (for C|�| > 0,
�N > 0)
d
dtyN(t) +
�N2|||f |||2⌫,N(t) 0.
where the modified instant energy is
yN(t) ⌘ 1
2
X
|↵|+|�|N
C|�|||@↵� f(t)||2 + |||[E, B � B]|||2N(t).
To obtain a decay rate in t is to bound the instant energy yN(t) in terms of the
dissipation rate |||f |||2⌫,N(t). Since the collision frequency ⌫(v) is bounded from below
in our hard-sphere interaction, part of yN(t) is clearly bounded by the dissipation
rate because
(3.6)1
2
X
|↵|+|�|N
C|�|||@↵� f(t)||2 C|||f |||2⌫,N(t).
On the other hand, to estimate the field component [E, B� B] of yN(t), Lemma 9 in
[39] only implies
(3.7) |||[E, B � B]|||2N�1
(t) C|||f |||2⌫,N(t).
No estimates for the highest N-th derivatives of E and B are available. Hence for
fixed N, |||f(t)|||2⌫,N is weaker than the total instant energy yN(t).
The key is to use interpolation to get a lower bound for the N-th order derivatives
@↵E and @↵B with the bound of higher energy |||f |||2N+k(t), for some k large. Let
@↵ = @↵0t @↵0
x and |↵| = N. For purely spatial derivaitves with ↵0
= 0 and |↵0| = N, a
simple interpolation between spatial Sobolev spaces HN�1
x and HN+kx yields
||@↵E||2 + ||@↵B||2 = ||@↵0
x E||2 + ||@↵0
x {B � B}||2
C||[E, B � B]||2k/(k+1)
HN�1x
⇥ ||[E, B � B]||2/(k+1)
HN+k
x
C||[E, B � B]||2k/(k+1)
N�1
||[E, B � B]||2/(k+1)
N+k .
By our assumption and Theorem 1 in [39]
||[E, B � B]||2N+k(t) CN+kEN+k(0) < 1.
84
We thus conclude by (3.7) that
||@↵E||2 + ||@↵B||2 C (CN+kEN+k(0))1/(k+1) ||[E, B � B]||2k/(k+1)
N�1
C (EN+k(0))1/(k+1) |||f |||2k/(k+1)
⌫,N .(3.8)
Now for the general case with ↵0
6= 0, we take @↵0�1
t @↵0x of the Maxwell system
(3.2) to get
@↵0t @↵0
x E = @↵0�1
t @↵0
x rx ⇥B �Z p
µ@↵0�1
t @↵0
x {f+
� f�}dv,
@↵0t @↵0
x B = �@↵0�1
t @↵0
x rx ⇥ E.
Since ||R p
µ@↵0�1
t @↵0x {f+
� f�}dv||2L2(T3
)
is clearly bounded by C|||f(t)|||2N or equiv-
alently CyN(t), which is bounded by
C (EN+k(0))1/(k+1) |||f |||2k/(k+1)
⌫,N .
We therefore deduce that (3.8) is valid for general ↵ with ↵0
� 1 via a simple induction
of ↵0
, starting from ↵0
= 1. Moreover, for the full dissipation rate
�N2|||f |||2⌫,N � CN,k (EN+k(0))
�1/k y{k+1}/kN .
It follows that
(3.9)dyN
dt+ CN,k (EN+k(0))
�1/k y{k+1}/kN 0,
and y0N(t)(yN(t))�1�1/k �CN,k (EN+k(0))�1/k . Integrating this over [0, t], we have
k{yN(0)}�1/k � k{yN(t)}�1/k � (EN+k(0))�1/k CN,kt.
Hence
{yN(t)}�1/k � tCN,k
k(EN+k(0))
�1/k + {yN(0)}�1/k,
since we can assume yN(0) CN,kEN+k(0), our theorem thus follows. ⇤85
3.3. Relativistic Landau-Maxwell System
The relativistic Landau-Maxwell system is the most fundamental and complete
model for describing the dynamics of a dilute collisional cold plasma in which par-
ticles interact through Coulombic collisions and through their self-consistent elec-
tromagnetic field. For notational simplicity, we consider the following normalized
Landau-Maxwell system
@tF+
+p
p0
·rxF+
+
✓
E +p
p0
⇥B
◆
·rpF+
= C(F+
, F+
) + C(F+
, F�)
(3.10)
@tF� +p
p0
·rxF� �✓
E +p
p0
⇥B
◆
·rpF� = C(F�, F�) + C(F�, F+
),
with initial condition F±(0, x, p) = F0,±(x, p). Here F±(t, x, p) � 0 are the spatially
periodic number density functions for ions (+) and electrons (�) at time t � 0,
position x = (x1
, x2
, x3
) 2 T3 and momentum p = (p1
, p2
, p3
) 2 R3. The energy of a
particle is given by p0
=p
1 + |p|2.
To completely describe a dilute plasma, the electromagnetic field E(t, x) and
B(t, x) is coupled with F±(t, x, p) through the celebrated Maxwell system (3.2) where
the charge density ⇢ and the current density J are given by
(3.11) ⇢ =
Z
R3
{F+
� F�} dp, J =
Z
R3
p
p0
{F+
� F�}dp.
The collision between particles is modelled by the relativistic Landau collision
operator C, and its normalized form is given by
C(g, h)(p) ⌘ rp ·⇢
Z
R3
�(P, P 0){rpg(p)h(p0)� g(p)rp0h(p0)}dp0�
,
where the four-vectors are P = (p0
, p1,p2,p3
) and P 0 = (p00
, p01
, p02
, p03
). Morover,
�(P, P 0) ⌘ ⇤(P, P 0)S(P, P 0) with ⇤(P, P 0) ⌘ (P ·P 0)2p0p00
{(P · P 0)2 � 1}�3/2 and
S(P, P 0) ⌘�
(P · P 0)2 � 1
I3
+ {(P · P 0)� 1} (p⌦ p0)� (p� p0)⌦ (p� p0).
86
Here the Lorentz inner product is P ·P 0 = p0
p00
�p·p0. Let �ij(p) =R
�ij(P, P 0)J(p0)dp0,
where J(p) ⌘ e�p0 is a relativistic Maxwellian. We define
||g||2� ⌘X
i,j
Z
T3⇥R3
⇢
2�ij@pj
g@pi
g +�ij
2
pi
p0
pj
p0
g2
�
dxdp.
See [62] for more details. We write F± ⌘ J +p
Jf± and study the decay rate for f±, E
and B. Define |||f |||2N(t) and |||[E, B]|||2N(t) as in (3.4) with @↵� ⌘ @↵0
t @↵1
x1@↵2
x2@↵3
x3@�1
p1@�2
p2@�3
p3
and dv = dp. Then the initial data is again measured by (3.5), but the dissipation
rate is given by
|||f |||2N,�(t) ⌘X
|↵|+|�|N
||@↵� f
+
||2� +X
|↵|+|�|N
||@↵� f�||2�.
We study the decay of solutions near F± ⌘ J(p), E ⌘ 0 and B ⌘ B = a constant.
Theorem 3.2. Fix N � 4, k � 1 and choose initial data [F0,±, E
0
, B0
] which sat-
isfies the assumptions of Theorem 1 in [62] for N +k (therefore also for N) including
EN+k(0) ✏N+k, ✏N+k > 0.
Let [F± = J +p
Jf±, E, B] be the unique solution to (3.10) and (3.2) with (3.11)
from Theorem 1 of [62], then there exists CN,k > 0 such that
|||f |||2N(t) + |||[E, B � B]|||2N(t) CN,kEN+k(0)
⇢
1 +t
k
��k
,
where B = 1
|T3|R
T3 B(t, x)dx is constant in time [62].
The proof is exactly as in the case for the Vlasov-Maxwell-Boltzmann system
because of two facts: C||g||� � ||g|| (Lemma 2 in [62]) and from Lemma 12 in [62]
|||[E, B]|||N�1
(t) C|||f |||N,�(t).
3.4. Soft Potentials
The Boltzmann equation with a soft potential is
@tF + v ·rxF = Q(F, F ), F (0, x, v) = F0
(x, v),
87
where F (t, x, v) is the spatially periodic distribution function for the particles at time
t � 0, position x = (x1
, x2
, x3
) 2 T3 and velocity v = (v1
, v2
, v3
) 2 R3. The l.h.s. of
this equation models the transport of particles and the operator on the r.h.s. models
the e↵ect of collisions on the transport:
Q(F, F ) ⌘Z
R3⇥S2
|u� v|�B(✓){F (u0)F (v0)� F (u)F (v)}dud!.
The exponent in the collision operator is � = 1� 4
swith 1 < s < 4 so that �3 < � < 0
(soft potentials), and B(✓) satisfies the Grad angular cuto↵ assumption: 0 < B(✓)
C| cos ✓|.
Notation: We define the collision frequency ⌫(v) ⌘R
|v�u|�µ(u)B(✓)dud!, and
||g||2⌫ =
Z
T3⇥R3
⌫(v)g2dxdv.
Notice that ⌫(v) ⇠ C|v|� as |v|!1. Define a weight function in v by w = w�(v) =
[1 + |v|]� with �3 < � < 0 and denote a weighted L2 norm as
||g||2` ⌘Z
T3⇥R3
w2`|g|2dxdv, ||g||2⌫,` ⌘Z
T3⇥R3
w2`⌫g2dxdv.
Let @↵� ⌘ @↵0
t @↵1
x1@↵2
x2@↵3
x3@�1
v1@�2
v2@�3
v3which includes temporal derivatives. For fixed
N � 4, we define the family of norms depending on ` as
|||f |||2`(t) ⌘X
|↵|+|�|N
||@↵� f(t)||2|�|�`,
|||f |||2⌫,`(t) ⌘X
|↵|+|�|N
||@↵� f(t)||2⌫,|�|�`.
Since � < 0, these norms include higher powers of v for larger ` � 0. We denote the
perturbation f such that F (t, x, v) = µ(v) +p
µ(v)f(t, x, v). Then the initial value
problem can be written as
(3.12) [@t + v ·rx]f + Lf = �[f, f ],
where L is the linear part of the collision operator Q and � is the non-linear part.
Define the instant energy as E`(f(t)) ⌘ 1
2
|||f |||2`(t), with its initial counterpart E`(0) ⌘1
2
|||f0
|||2` . At t = 0 we define the temporal derivatives naturally through (3.12).
88
Theorem 3.3. Fix �3 < � < 0 and fix integers N � 8, ` � 0, k � 0. Choose
initial data F0
(x, v) which satisfies the assumptions of Theorem 1 in [37]. Further-
more, there are constants C` > 0 and ✏` > 0, such that if E`(0) < ✏` then there exists
a unique global solution with F (t, x, v) = µ + µ1/2f(t, x, v) � 0, where equivalently f
satisfies (3.12) and
(3.13) sup0s1
E`(f(s)) C`E`(0).
Moreover if, E`+k(0) < ✏`+k, then the unique global solution to the Boltzmann equation
(3.12) satisfies
|||f |||2`(t) C`,kE`+k(0)
✓
1 +t
k
◆�k
.
Remark 3.1. The existence part of both this theorem and the next were proven for
` = 0 and with no temporal derivatives in [37] and [38]. But the method was improved
in [39]. Since the proof is very similar to those in [37–39,62], we will only sketch it.
Proof. 1. BASIC ESTIMATES: Our first goal is to show all the basic estimates
in [37] are valid, with little changes in the proofs, for new norms with (harmless)
temporal derivatives and an additional powers of the weight factor w. In what follows,
we shall only pin point these minor chanegs. Lemma 1 in [37] holds for any ✓ < 0
with the same proof except we replace Eq. (19) by
w2✓(v) w2✓(|v|+ |u|) Cw2✓(|v0|+ |u0|) Cw2✓(v0)w2✓(u0).
One factor on the r.h.s. is then controlled by eitherp
µ(v0) orp
µ(u0) in Eq. (18)
of [37]. Lemma 2 in [37] holds for ✓ < 0 as well and the only di↵erence is in Eqs.
(48)-(50) of [37] we should use w2✓(v) Cw2✓(v + u||)w2✓(u||) instead of w2✓(v) 1
(which is used for the case ✓ � 0). Not only Lemma 4 in [37] is valid for ✓ < 0
with the same proof, it is also valid for the case � = 0 as well. In fact, for ✓ < 0,
by Lemma 2 in [37], we split K = Kc + Ks where�
�
�
w2✓Ks@↵f, @↵f�
�
� 1
2
k@↵fk2
⌫,✓
by a trivial extension of Eq. (21) in [37] to the case ✓ < 0. Finally, using the
definition of Kc in Eqs. (24) and (40) as well as the Cauchy-Schwartz inequality we
have�
�
�
w�2✓Kc@↵f, @↵f�
�
� Ck@↵fk2
⌫ for some constant C > 0 which completes the
89
case for � = 0 in Lemma 4 in [37]. Furthermore, by splitting L = ⌫ + K, we deduce
that for some C > 0 and ` � 0,
(3.14)�
w�2`L@↵f, @↵f�
� 1
2k@↵fk2
⌫,�` � Ck@↵fk2
⌫ .
Theorem 3 in [37] is valid for any weight function w2✓�2` instead of w2✓ with the
same proof. Now Lemma 7 and the local well-posedness Theorem 4 in [37] are valid
with the new norms with the same proof.
2. POSITIVITY OF L : Instead of Theorem 2 in [37], we show the following
instantaneous positivity for the linearized Boltzmann operator L: Fix ` � 0. Let
f(t, x, v) be a (local) classical solution to the Boltzmann equation with a soft potential
with the new norm. There exists M0
and �0
= �0
(M0
) > 0 such that if E`(f(t)) M0
,
then
(3.15)X
|↵|N
hL@↵f(t), @↵f(t)i � �0
X
|↵|N
k@↵f(t)k2
⌫ .
The proof of (3.15) follows exactly as in section 4 in [39] in a much simplier fashion
with no electromagnetic fields everywhere. Both Lemma 5 and Lemma 6 in [39] holds
with E = B = 0. Lemma 7 (with J = 0) in [39] holds with new L for soft potentials
and with ||@�f || replaced by ||@�f ||⌫ , thanks to Lemma 1 in [37]. Lemma 8 (with
E = B = 0) in [39] is valid for new � for the soft potentials with ||@�f || replaced by
||@�f ||⌫ , thanks to Lemma 6 in [37]. The rest of the proof is the same (much simpler!)
as the proof of Theorem 3 starting at p.620 in [39].
Remark. We remark that the assumption E`(f(t)) M0
in this section can
be reduced to not include any velocity derivatives. However this requires a slight
improvement of estimates found in [37]. For instance, one can prove that for |↵1
| +
|↵2
| N and �1
(v) a smooth function such thatP
|�|2
|@��1
| Cµ1/4(v), then
kh�[@↵1g, @↵2g]�1
ik CX
|↵|N
||@↵g||⌫X
|↵|N
||@↵g||.
Further, if |↵| N , khL[@↵g],�1
ik C||@↵g||.
3. ENERGY ESTIMATE. Fix an integer ` � 0. Let f be the unique local solution
which satisfies the small amplitude assumption E`(f(t)) M0
. We now show that for
90
any given 0 m N , |�| m, there are constants C|�|,¯` > 0, C⇤m,` > 0 and �m,` > 0
such that
X
|�|m,0¯``,|↵|+|�|N
✓
1
2
d
dtC|�|,¯`||@↵
� f(t)||2|�|�¯` + �m,`||@↵� f(t)||2⌫,|�|�¯`
◆
C⇤m,`
p
E`(t)|||f |||2⌫,`(t).(3.16)
We derive (3.16) in three steps.
We first consider pure spatial and temporal derivatives. Taking pure @↵ derivatives
of (3.12) we obtain
(3.17) [@t + v ·rx]@↵f + L@↵f = @↵�[f, f ].
Take the inner product of this with @↵f . We apply (3.15) and Theorem 3 in [37]
with the new norms to obtain for some constant C > 0,
X
|�|N
✓
1
2
d
dt||@↵f(t)||2 + �
0
||@↵f(t)||2⌫◆
Cp
E0
(t)|||f |||2⌫(t).
Next, we consider pure spatial and temporal derivatives with weights. We prove
the result for ` > 0 by a simple induction starting at ` = 0, assume (3.16) is valid for
0 `0 < `. Take the inner product of (3.17) with w�2`@↵f . Then for some constant
C > 0, we obtain the bound
X
|↵|N
✓
1
2
d
dt||@↵f(t)||2�` +
�
w�2`L@↵f, @↵f�
◆
Cp
E`(t)|||f |||2⌫,`(t).
Plug in the lower bound (3.14) and add the cases 0 `0 < ` multiplied by a large
constant to establish (3.16) for |�| = 0 and ` � 0.
Finally we prove the general case by an induction over the order of v derivatives.
Assume (3.16) is valid for |�| = m. For |�| = m + 1, we take @↵� of (3.12) to obtain
(3.18) [@t + v ·rx]@↵� f + @�L@↵f = �
X
|�1|=1
✓
�
�1
◆
@�1v ·rx@↵���1
f + @↵��[f, f ].
91
Take the inner product of this with w2|�|�2`@↵f . Using Lemma 4 of [37] with ✓ =
2|�|� 2` we get
�
w2|�|�2`@�{L@↵f}, @↵� f�
� ||@↵� f ||2⌫,|�|�` � ⌘
X
|¯�||�|||@↵
¯� f ||2⌫,|¯�|�` � C⌘||@↵f ||2⌫,�`.
Using Cauchy’s inequality for ⌘ > 0, since |�1
| = 1 we have
�
�
�
w2|�|�2`@�1v ·rx@↵���1
f, @↵� f�
�
� ⌘||@↵� f(t)||2⌫,|�|�` + C⌘||rx@
↵���1
f ||2⌫,|���1|�`,
which is found on p.336 of [37]. We use these two inequalities to conclude (3.16) as
well as global existence and (3.13) exactly as in section 5 in [39,62].
5. DECAY RATE: We let m = N in (3.16) and define
y`(t) ⌘X
0¯``,|↵|+|�|N
C|�|,¯`||@↵� f(t)||2|�|�¯`.
Notice that y`(t) is equivalent to |||f |||2`(t). By (3.13) for su�ciently small E`(0), there
exists C > 0 such thatd
dty`(t) + C|||f |||2⌫,`(t) 0.
Clearly for fixed `, the dissipation rate |||f |||2⌫,` is weaker than the instant energy
y`(t) since kfk1/2
is equivalent to kfk⌫ . We shall interpolate with the stronger norm
y`+k(t). Fix `, k � 0. A simple intepolation for w�2` between the weight functions
w�2`+2 and w�2`�2k yields
|||f |||` |||f |||k/(k+1)
`�1
|||f |||1/(k+1)
`+k C|||f |||k/(k+1)
`�1
⇥ (C`+kE`+k(0))1/2(k+1) .
But from the definition of w(v) and the fact ⌫(v) Cw(v) we have
|||f |||2`�1/2
C|||f |||2⌫,`.
We thus have
y` C|||f |||2` CE1/(k+1)
`+k (0)|||f |||2k/(k+1)
`�1
CE1/(k+1)
`+k (0)|||f |||2k/k+1
⌫,` ,
and for some C`,k > 0,
d
dty` + C`,kE�1/k
`+k (0)y{k+1}/k` 0.
92
Now the theorem follows from the analysis of (3.9). ⇤
3.5. The Classical Landau Equation
The Landau equation has the same form as the Boltzmann equation except
Q(F, F ) = rv ·⇢
Z
R3
�(v � v0)[F (v0)rvF (v)� F (v)rv0F (v0)]dv0�
,
where �ij(v) = 1
|v|
n
�ij � vi
vj
|v|2o
. Throught out this section we use the weight function
w1
(v) ⌘ {1 + |v|2}�1/2.
In the classical case, we define �ij(v) ⌘R
�ij(v � v0)µ(v0)dv0 and
||g||2�,✓ ⌘X
i,j
Z
T3⇥R3
w2✓�ij{@vi
g@vj
g + vivjg2}dxdv.
We again define @↵� including temporal derivativess. Fix N � 4 and ` � 0 and define
|||f |||2`(t) ⌘P
|↵|+|�|N ||w2(|�|�`)@↵� f(t)||2. The Landau dissipation rate is
|||f |||2�,`(t) ⌘X
|↵|+|�|N
||@↵� f(t)||2�,|�|�`,
The standard perturbation f(t, x, v) to µ is F (t, x, v) = µ(v)+p
µ(v)f(t, x, v). The
instant energy is E`(f(t)) ⌘ 1
2
|||f |||2`(t) with its initial counterpart E`(0) ⌘ 1
2
|||f0
|||2` .
Theorem 3.4. Fix integers N � 8 and ` � 0. Assume that F0
(x, v) = µ +
µ1/2f0
(x, v) satisfies the assumptions of Theorem 1 in [38]. There are constants C` > 0
and ✏` > 0 such that if E`(f0
) < ✏` then there exists a unique global solution F (t, x, v)
to the classical Landau equation such that F (t, x, v) = µ + µ1/2f(t, x, v) � 0, and
sup0s1
E`(f(s)) C`E`(0).
Furthermore if E`+k(f0
) < ✏`+k for some integer k > 0 then the unique global solution
f(t, x, v) satisfies
|||f |||2`(t) C`,kE`+k(0)
✓
1 +t
k
◆�k
.
93
Proof. We follow the same procedure as in the case for soft potentials. First,
we check that basic estimates in [38] are valid with little changes in the proofs for
new norms with (harmless) temporal derivatives as well as new weights w = w�2`1
.
We first notice that Lemma 6 in [38] is valid for |�| = 0 as well. The result for K is
Lemma 5 in [38]. Therefore we focus on A. Lemma 3 in [38] implies
|@vj
{@vi
w�2`1
�ij}|+ |@vj
{w�2`1
@vi
�ij}| Cw2`1
[1 + |v|]�2.
Now both in second and the fourth term in Eq. (31) in [38] with � = 0, we write
@jff = 1
2
@jf 2 and integate by parts to get a upper bound of
(3.19) CX
i,j
Z
T3⇥R3
w�2`1
[1 + |v|]�2 |f |2 dxdv
Since ||f ||�,�` dominates ||f ||�`+1/2
, (3.19) is therefore bounded by ⌘||f ||�,�` for large
v for any small number ⌘. This concludes the proof of Lemma 6 in [38] for the new
norm. All the other Lemmas in sections 1-3 are valid for the new norms with identical
proofs, which lead to local well-posedness Theorem 4 in [38] for the new norm.
Now we follow exactly the proof for the soft potential, simply replacing ||| · |||⌫by the corresponding Landau dissipation rate ||| · |||� and the linearized Boltzmann
operator by the linearized Landau operator. To establish the positivity (3.15) with
Landau dissipation, we notice that Lg = ��(g,p
µ)� �(p
µ, g). By Lemma 7 in [38]
for the new norms, we deduce that both Lemma 7 and Lemma 8 in [39] for Landau
dissipation rate ||@�f ||2�, instead of ||@�f ||2. The rest of the proof is identical. ⇤
94
CHAPTER 4
Exponential Decay for Soft Potentials near Maxwellian
Abstract. Consider both soft potentials with angular cuto↵ and Landau collision
kernels in the Boltzmann theory inside a periodic box. We prove that any smooth
perturbation near a given Maxwellian approaches to zero at the rate of e��tp for some
� > 0 and 0 < p < 1. Our method is based on an unified energy estimate with
appropriate exponential velocity weight. Our results extend the classical Caflisch
result [9] to the case of very soft potential and Coulomb interactions, and also improve
the recent “almost exponential” decay results by [21,60]. A version of this result will
appear as [61].
4.1. Introduction
4.2. Introduction
In this article, we are concerned with soft potentials and Landau collision kernels
in the Boltzmann theory for dynamics of dilute particles in a periodic box. Recall
the Boltzmann equation as
(4.1) @tF + v ·rxF = Q(F, F ), F (0, x, v) = F0
(x, v),
where F (t, x, v) is the spatially periodic distribution function for the particles at time
t � 0, position x = (x1
, x2
, x3
) 2 T3 and velocity v = (v1
, v2
, v3
) 2 R3. The l.h.s. of
this equation models the transport of particles and the operator on the r.h.s. models
the e↵ect of collisions on the transport:
Q(F, G) ⌘Z
R3⇥S2
|u� v|�B(✓){F (u0)G(v0)� F (u)G(v)}dud!.
95
Here F (u) = F (t, x, u) etc. The exponent is � = 1� 4
swith 1 < s < 4; we assume
�3 < � < 0,
(soft potentials) and B(✓) satisfies the Grad angular cuto↵ assumption:
(4.2) 0 < B(✓) C| cos ✓|.
Moreover, the post-collisional velocities satisfy
v0 = v + [(u� v) · !]!, u0 = u� [(u� v) · !]!,(4.3)
|u|2 + |v|2 = |u0|2 + |v0|2,(4.4)
And ✓ is defined by cos ✓ = [u� v] · !/|u� v|.
On the other hand, the Landau equation is formally obtained in a singular limit
of the Boltzmann equation. It can also be written as (4.1) but the collision operator
is given by
Q(F, G) = rv ·⇢
Z
R3
�(v � u)[F (u)rvG(v)�G(v)ruF (u)]du
�
=3
X
i,j=1
@i
Z
R3
�ij(v � u)[F (u)@jG(v)�G(v)@jF (u)]du,
where @i = @vi
etc. The non-negative matrix � is given by
�ij(v) =
⇢
�ij �vivj
|v|2�
|v|2+�.
We assume soft potentials, which means �3 � < �2. The original Landau collision
operator with Coulombic interactions corresponds to � = �3.
Denote the steady state Maxwellian by
µ(v) = (2⇡)�3/2e�|v|2/2.
We perturb around the Maxwellian as
F (t, x, v) = µ(v) +p
µ(v)f(t, x, v).
Then the initial value problem (4.1) can be rewritten as
(4.5) [@t + v ·rx]f + Lf = �[f, f ], f(0, x, v) = f0
(x, v),
96
where L is the linear part of the collision operator, Q, and � is the non-linear part.
For the Boltzmann equation, the standard linear operator [30] is
(4.6) Lg = ⌫(v)g �Kg,
where the collision frequency is
(4.7) ⌫(v) =
Z
B(✓)|v � u|�µ(u)dud!.
The operators K and �, in the Boltzmann case, are defined in (4.17) and (4.46).
For the Landau equation, the linear operator [38] is
(4.8) Lg = �Ag �Kg,
with A, K and � defined in (4.44), (4.45) and (4.46). The Landau collision frequency
is
(4.9) �ij(v) = �ij ⇤ µ =
Z
R3
�ij(v � u)µ(u)du.
We remark that �ij(v) is a positive self-adjoint matrix [16].
Notation: Let h·, ·i denote the standard L2(R3) inner product. We also use (·, ·)
to denote the standard L2(T3 ⇥R3) inner product with corresponding L2 norm k · k.
Define a weight function in v by
(4.10) w = w(`,#)(v) ⌘ (1 + |v|2)⌧`/2 exp⇣q
4(1 + |v|2)#
2
⌘
.
Here ⌧ < 0, ` 2 R, 0 < q and 0 # 2. Denote weighted L2 norms as
|g|2`,# ⌘Z
R3
w2(`,#)|g|2dv, ||g||2`,# ⌘Z
T3
|g|2`,#dx.
For the Boltzmann equation, define the weighted dissipation norm as
|g|2⌫,`,# ⌘Z
R3
w2(`,#)⌫(v)|g(v)|2dv,(4.11)
||g||2⌫,`,# ⌘Z
T3
|g|2⌫,`,#dx.
97
For the Landau equation, define the weighted dissipation norm as
|g|2�,`,# ⌘3
X
i,j=1
Z
R3
w2(`,#)n
�ij@ig@jg + �ij vi
2
vj
2|g|2o
dv,(4.12)
||g||2�,`,# ⌘Z
T3
|g|2�,`,#dx.
Since our proof of decay does not depend upon any properties which are specific to
either dissipation norm, sometimes we unify the notation as ||g||D,`,#, which denotes
either ||g||⌫,`,# or ||g||�,`,#. If # = 0 then we drop the index, e.g. ||g||D,`,0 = ||g||D,`
and the same for the other norms.
Next define a high order derivative
@↵� ⌘ @↵0
t @↵1
x1@↵2
x2@↵3
x3@�1
v1@�2
v2@�3
v3
where ↵ = [↵0,↵1,↵2,↵3] is the multi-index related to the space-time derivative
and � = [�1, �2, �3] is the multi-index related to the velocity derivatives. If each
component of � is not greater than that of �1
’s, we denote by � �1
; � < �1
means
� �1
and |�| < |�1
|. We also denote
0
@
�
�1
1
A by C�1� .
Fix N � 8 and l � 0. An “Instant Energy functional” satisfies
1
CEl,#(g)(t)
X
|↵|+|�|N
||@↵� g(t)||2|�|�l,# CEl,#(g)(t).
If g is independent of t � 0, then the temporal derivatives are defined through equa-
tion (4.5). Further, the “Dissipation Rate” is given by
Dl,#(g)(t) ⌘X
|↵|+|�|N
||@↵� g(t)||2D,|�|�l,#.
We will also write El,0(g)(t) = El(g)(t) and Dl,0(g)(t) = Dl(g)(t). We note from (4.10)
that for l > 0 these norms contain a polynomial factor (1 + |v|2)�⌧ l/2. The weight
factor (1+ |v|2)⌧ |�|/2 (dependant on the number of velocity derivatives) is designed to
control the streaming term v ·rxf .
98
If initially F0
(x, v) = µ(v) +p
µ(v)f0
(x, v) has the same mass, momentum and
total energy as the Maxwellian µ, then formally for any t � 0 we have
(4.13)
Z
T3⇥R3
f(t)µ1/2 =
Z
T3⇥R3
vif(t)µ1/2 =
Z
T3⇥R3
|v|2f(t)µ1/2 = 0.
We are now ready to state the main result.
Theorem 4.1. Let N � 8, l � 0, 0 # 2 and 0 < q. If # = 2 then further
assume q < 1. Choose initial data F0
(x, v) = µ(v) +p
µ(v)f0
(x, v) such that f0
(x, v)
satisfies (4.13). In (4.10), for the Boltzmann case assume ⌧ � and for the Landau
case assume ⌧ �1.
Then there exists an instant energy functional El,#(f)(t) such that if El,#(f0
) is
su�ciently small, then the unique global solution to (4.1) in both the Boltzmann case
and the Landau case satisfies
(4.14)d
dtEl,#(f)(t) +Dl,#(f)(t) 0.
In particular,
(4.15) sup0s1
El,#(f)(s) El,#(f0
).
Moreover, if # > 0, then there exists � > 0 such that
El(f)(t) Ce��tpEl,#(f0
),
where in the Boltzmann case
p = p(#, �) =#
#� �,
and in the Landau case
p = p(#, �) =#
#� (2 + �).
In the second authors papers [37, 38], (4.14) was established with ⌧ = � in the
Boltzmann case, ⌧ = 2 + � in the Landau case and # = l = 0. We extended (4.14)
to the case l � 0 in [60]. There we used (4.14) and (4.15) to establish the following
theorem via direct interpolation for # = 0.
99
Theorem 4.2. Assume everything from Theorem 4.1 and fix k > 0. In addition,
if El+k,#(f0
) is su�ciently small then
El,#(f)(t) Cl,k
✓
1 +t
k
◆�k
El+k,#(f0
).
For # > 0, the proof of Theorem 4.2 is exactly the same as in [60]. Also in [60],
for the Landau case we assumed Coulomb interactions (� = �3). Still the proof of
Theorem 4.2 in the Landau case for ⌧ �1 and �3 � < �2 is exactly the same.
However, we remark that for ⌧ < 2 + � the interpolations used to prove Theorem
4.2 are not optimal. The polynomial decay rate can be improved as ⌧ grows smaller
than � by using tighter interpolations. But this is somewhat superficial because as ⌧
grows smaller we are implicitly using a larger weight in our norms.
The main di�culty in proving any kind of decay for soft potentials is caused by
the lack of a spectral gap for the linear operator (4.6) and (4.8). In the Boltzmann
case, the dominant part of the linear operator (4.6) is of the form
(4.16)1
C(1 + |v|2)�/2 ⌫(v) C(1 + |v|2)�/2, C > 0.
From another point of view, at high velocities the dissipation is much weaker than
the instant energy. However, Theorem 4.1 and Theorem 4.2 show that given explicit
control over f(t, x, v) at high velocities, no matter how weak, we can obtain a precise
decay rate. On the other hand, we believe it very unlikely that existence of solutions
can be established in this setting with a weight bigger than (4.10) with # = 2. From
this point of view, Theorem 4.1 and Theorem 4.2 together form a rather satisfactory
theory of convergence rates to Maxwellian for soft potentials and Landau operators,
in a close to equilibrium context.
The constants in our estimates are certainly not optimal or explicit in all cases.
However, p = ##��
comes from the following simplification of the Boltzmann equation
[8]:
@tf(t, |v|) + |v|�f(t, |v|) = 0, �3 < � < 0, |v| > 0.
100
Consider initial data with rapid decay as required by our norms
f(0, |v|) = e�c|v|# , c > 0, 0 < # 2.
Then the solution to this system is exactly
f(t, |v|) = e�c|v|#�t|v|�
By splitting into {|v| � tp/#} and {|v| < tp/#} we have
c0
e�c1tp Z
|v|>0
|f(t, |v|)|2d|v| c2
e�c3tp ,
with ci > 0 (i = 0, 1, 2, 3).
The study of trend to Maxwellians is important in kinetic theory both from phys-
ical and mathematical standpoints. In a periodic box, it was Ukai [65] who obtained
exponential convergence (with p = 1), and hence constructed the first global in time
solutions in the spatially inhomogeneous Boltzmann theory. He treated the case of
a cuto↵ hard potential. In 1980, Caflisch [8, 9] established exponential decay (with
the same p(2, �)) as well as global in time solutions for the Boltzmann equation with
potentials which are not too soft (�1 < � < 0). About the same time, in the whole
space setting, also for cuto↵ soft potentials with � > �1, Ukai and Asano [66] ob-
tained the rate O(t�↵) with 0 < ↵ < 1; their optimal case in R3 yields ↵ = 3/4. In
these early investigations, a su�ciently fast time decay of the linearized Boltzmann
equation around a Maxwellian played the crucial role in bootstrapping to the full
nonlinear dynamics. For the soft potentials, such linear decay estimates can be very
di�cult and delicate. It thus has been a longstanding open problem to study the
decay property as well as to construct global in time smooth solutions for very soft
potentials with � near �3.
Recently, a nonlinear energy method for constructing global solutions was de-
veloped by the second author to avoid using the linear decay. Indeed, by showing
the linearized collision operator was always positive definite along the full nonlinear
dynamics, global in time smooth solutions near Maxwellians were constructed for
all cuto↵ soft potentials of �3 < � < 0 [37], even for the Landau equation with
101
Coulomb interaction [38]. However, the time decay of such solutions was left open.
See [36,39,40,62] for more applications of such a method.
From a completely di↵erent approach, Desvillettes and Villani [21] have recently
developed a framework to study the trend to Maxwellians for general smooth solu-
tions, not necessarily near any Maxwellian. As an application, their method leads to
the almost exponential decay rates (i.e., faster than any given polynomial) for smooth
solutions constructed earlier by the second author for all cuto↵ soft potentials and
the Landau equation.
Inspired by such a striking result, in [60], we re-examined and improved the energy
method to give a more direct proof of such almost exponential decay in the close to
Maxwellian setting. We introduced a family of polynomial velocity weight functions
and used some simple interpolation techniques. It is interesting to note that our decay
estimate is a consequence of the weighted energy estimate for the global solution, not
the other way around as in earlier methods [8, 9, 65, 66]. And it is clear from our
analysis that a stronger velocity weight yields faster time decay.
It is thus very natural to try to use exponential velocity weight functions to get
exponential time decay, which is the main purpose of our current investigation. The
key is to show that the new energy with an exponential velocity weight is bounded
for all time. In order to carry out such an energy estimate, we follow the general
framework and strategy in [37], [38], [60]. However, many new analytical di�culties
arise and we have to develop new techniques accordingly. The main di�culty lies in
the estimates for linearized collision operators around a Maxwellians. The presence
of the exponential weight factor exp{ q4
(1 + |v|2)#/2} in (4.10) requires much more
precise estimates at almost all levels. In the case of a cuto↵ soft potential, a careful
application of the Caflisch estimate (Lemma 4.1) is combined with the splitting trick
in [37] to treat the very soft potential of �3 < � �1. Furthermore, we take a close
look at the variables (4.21) in the Hilbert-Schmidt form for K to estimate the trickiest
terms in Lemma 4.2. On the other hand, in the Landau case, an extra v factor from
the derivative of the weight exp{ q4
(1+ |v|2)#/2} creates the most challenging di�culty
102
to close the estimate in the same norm. We have to use di↵erent weight functions,
that appeared in the norm (4.12), very precisely to balance between the derivative
part �ij@ig@jg and the no derivative part �ij vi
2
vj
2
|g|2 in Lemma 4.8. Thus ⌧ �1
is assumed. Moreover, in Lemma 4.9, we have to introduce a new splitting of the
linear Landau operator in the # = 2 case where q < 1 is crucially used. Although we
have quoted some estimates in previous papers, we have made an e↵ort to provide
complete, self-consistent proofs throughout the article. We also have made more
comments between the proofs to help the readers.
The paper is organized as follows. In Section 4.3, we establish the estimates
with the exponential weight (4.10) for the Boltzmann equation. In Section 4.4, we
establish estimates with weight (4.10) for the Landau equation. Finally, in Section
4.5 we establish the crucial energy estimate uniformly for both cases. Some details
are exactly the same as in [37], [38] or [60]. We will sketch these details which can be
found elsewhere. Finally, we prove exponential decay in Section 4.6 using the global
bound (4.15).
4.3. Boltzmann Estimates
In this section, we will prove the basic estimates used to obtain global existence
of solutions with an exponential weight in the Boltzmann case. These estimates are
similar to those in [37], but here the exponential weight which was not present earlier
forces us to modify the proofs and some of the estimates. We will use the classical
soft potential estimate of Caflisch [8] (Lemma 4.1) with v derivatives to estimate the
linear operator (Lemma 4.2). We discuss the new features of each proof after the
statement of each Lemma.
103
Recall K and � from (4.6) and (4.5). K = K2
�K1
is defined as [30,34]:
[K1
g](v) =
Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)µ1/2(v)g(u)dud!,(4.17)
[K2
g](v) =
Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)µ1/2(u0)g(v0)dud!
+
Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)µ1/2(v0)g(u0)dud!.
(4.18)
Consider a smooth cuto↵ function 0 �m 1 such that (for m > 0)
(4.19) �m(s) ⌘ 1, for s � 2m; �m(s) ⌘ 0, for s m.
Then define �m = 1� �m. Use �m to split K2
= K�2
+ K1��2
where
K�2
g ⌘Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)�m(|u� v|)µ1/2(u0)g(v0)dud!
+
Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)�m(|u� v|)µ1/2(v0)g(u0)dud!.
After removing the singularity at u = v, following the procedure in [30, 34] (see also
eqns. (35) and (36) in [37]), we can write
K�2
g =
Z
R3
k�2
(v, ⇠)g(v + ⇠)d⇠,
where
(4.20) k�2
(v, ⇠) ⌘ e�18 |⇠|2� 1
2 |⇣k|2
|⇠|p
⇡3/2
Z
R2
�m(p
|⇠|2 + |⇠?|2)(|⇠|2 + |⇠?|2)
1��
2
e�12 |⇠?+⇣?|2 B(✓)
| cos ✓|d⇠?.
The integration variables are d⇠? = d⇠1
?d⇠2
? but ⇠? = ⇠1
?⇠1 + ⇠2
?⇠2 2 R3 where
{⇠1, ⇠2, ⇠/|⇠|} is an orthonormal basis for R3. Also
(4.21) ⇣k =(v · ⇠)⇠|⇠|2 +
1
2⇠, ⇣? = v � (v · ⇠)⇠
|⇠|2 = (v · ⇠1)⇠1 + (v · ⇠2)⇠2.
It should be noted that, as in [37], we have removed the symmetry of the kernel
k�2
via the translation ⇠ ! v + ⇠. This formulation is well suited for taking high
order v-derivatives. Caflisch [8] proved Lemma 4.1 just below with no derivatives. In
contrast, we have already removed the singularity and this allows us to extend the
estimate from �1 < � < 0 to the full range �3 < � < 0. As in [37], we will see in
Lemma 4.2 that the singular part of K2
, e.g. K1��2
, has stronger decay.
104
Lemma 4.1. For any multi-index � and any 0 < s1
< s2
< 1, we have
|@�k�2
(v, ⇠)| Cexp
�
� s28
|⇠|2 � s12
|⇣|||2}�
|⇠|(1 + |v|+ |⇠ + v|)1��.
Here C > 0 will depend on s1
, s2
and �.
Proof. Fix 0 < s1
< s2
< 1. If |�| > 0, from (4.20) and (4.21) we have
@�k�2
(v, ⇠) =e�
18 |⇠|2
|⇠|p
⇡3/2
Z
R2
�m(p
|⇠|2 + |⇠?|2)(|⇠|2 + |⇠?|2)
1��
2
@�
⇣
e�12 |⇠?+⇣?|2� 1
2 |⇣k|2⌘ B(✓)
| cos ✓|d⇠?.
By a simple induction, for any 0 < q0 < 1, we have
�
�
�
@�
⇣
e�12 |⇠?+⇣?|2� 1
2 |⇣k|2⌘
�
�
�
C(|�|, q0)e� q
02 |⇠?+⇣?|2� q
02 |⇣k|2 .
Further restrict q0 > s1
. For |�| � 0, using the last display and (4.2) we have
|@�k�2
(v, ⇠)| Ce�
18 |⇠|2
|⇠|
Z
R2
�m(p
|⇠|2 + |⇠?|2)(|⇠|2 + |⇠?|2)
1��
2
e�q
02 |⇠?+⇣?|2� q
02 |⇣k|2d⇠?.
Change variables ⇠? ! ⇠? � ⇣? on the r.h.s. to obtain
Ce�
18 |⇠|2� q
02 |⇣k|2
|⇠|
Z
R2
�m(p
|⇠|2 + |⇠? � ⇣?|2)(|⇠|2 + |⇠? � ⇣?|2)
1��
2
e�q
02 |⇠?|2d⇠?.
Using (4.19), then, we have
(4.22) |@�k�2
(v, ⇠)| C(m)e�
18 |⇠|2� q
02 |⇣k|2
|⇠|
Z
R2
e�q
02 |⇠?|2d⇠?
(1 + |⇠|2 + |⇠? � ⇣?|2)1��
2
.
In the rest of the proof, we will refine this estimate via splitting.
Choose any q00 > 0 with q00 < s1
and then define ⌧⇤ =q
q0�s1
q0�q00 < 1. Split the
integration region as follows
{|⇠?| > ⌧⇤|⇣?|} [ {|⇠?| ⌧⇤|⇣?|}.
Further split the r.h.s. of (4.22) into k�,12
(v, ⇠)+k�,22
(v, ⇠) where k�,12
(v, ⇠) is restricted
to the region {|⇠?| > ⌧⇤|⇣?|}:
k�,12
(v, ⇠) ⌘ C(m)e�
18 |⇠|2� q
02 |⇣k|2
|⇠|
Z
|⇠?|>⌧⇤|⇣?|
e�q
02 |⇠?|2d⇠?
(1 + |⇠|2 + |⇠? � ⇣?|2)1��
2
.
For k�,12
(v, ⇠) we will observe exponential decay. And for k�,22
(v, ⇠) we can extract
from the denominator on the r.h.s. of (4.22) the exact decay stated in Lemma 4.1.
105
First consider k�,12
(v, ⇠). Since {|⇠?| > ⌧⇤|⇣?|} and q0 � q00 > 0 we have
�
�k�,12
(v, ⇠)�
� Ce�
18 |⇠|2� q
02 |⇣k|2
|⇠|
Z
{|⇠?|>⌧⇤|⇣?|}
e�q
002 |⇠?|2� q
0�q
002 |⇠?|2
(1 + |⇠|2 + |⇠? � ⇣?|2)1��
2
d⇠?
Ce�
18 |⇠|2� q
02 |⇣k|2
|⇠|
Z
{|⇠?|>⌧⇤|⇣?|}
e�q
002 |⇠?|2� q
0�s12 |⇣?|2
(1 + |⇠|2 + |⇠? � ⇣?|2)1��
2
d⇠?.
By (4.21),
|⇣k|2 + |⇣?|2 = |⇣k + ⇣?|2 = |v + ⇠/2|2.
Splitting q0 = s1
+ (q0 � s1
) we have
�
�k�,12
(v, ⇠)�
� C
|⇠|e� 1
8 |⇠|2�s12 |⇣k|2
Z
{|⇠?|>⌧⇤|⇣?|}
e�q
002 |⇠?|2� q
0�s12 (|⇣?|2+|⇣k|2)
(1 + |⇠|2 + |⇠? � ⇣?|2)1��
2
d⇠?
C
|⇠|e� 1
8 |⇠|2�s12 |⇣k|2� q
0�s12 |v+⇠/2|2 .
This will be more than enough decay. We expand
|v + ⇠/2|2 = |v|2 +1
4|⇠|2 + v · ⇠ =
1
4|v + ⇠|2 +
3
4|v|2 +
1
2v · ⇠
� 1
4|v + ⇠|2 +
3
4|v|2 � 1
4|v|2 � 1
4|⇠|2(4.23)
=1
4|v + ⇠|2 +
1
2|v|2 � 1
4|⇠|2.
Plug the last display into the one above it to obtain
�
�k�,12
(v, ⇠)�
� C
|⇠|e� 1
8 |⇠|2�s12 |⇣k|2e�
q
0�s18 |v+⇠|2� q
0�s14 |v|2+
q
0�s18 |⇠|2
=C
|⇠|e� s1+1�q
08 |⇠|2� s1
2 |⇣k|2e�q
0�s18 |v+⇠|2� q
0�s14 |v|2 .
Given s2
with s1
< s2
< 1, we can always choose q0, restricted by s1
< q0 < 1, such
that s2
= s1
+ 1� q0. This completes the estimate for k�,12
(v, ⇠).
106
On {|⇠?| ⌧⇤|⇣?|}, |⇣? � ⇠?| � |⇣?| � |⇠?| � (1 � ⌧⇤)|⇣?| (0 < ⌧⇤ < 1). Hence
(4.22) over this region is bounded as
�
�k�,22
(v, ⇠)�
� Ce�18 |⇠|2� q
02 |⇣k|2
|⇠| (1 + |⇠|2 + (1� ⌧⇤)2|⇣?|2)1��
2
Z
{|⇠?|⌧⇤|⇣?|}e�
q
02 |⇠?|2d⇠?
Ce�18 |⇠|2�
s12 |⇣k|2
|⇠|�
1 + |⇠|2 + |⇣?|2 + |⇣k|2�
1��
2
=Ce�
18 |⇠|2�
s12 |⇣k|2
|⇠| (1 + |⇠|2 + |v + ⇠/2|2)1��
2
,
where we used s1
< q0 to go from the first line to the second. Now plug (4.23) into
the last display to complete the estimate. ⇤
Next we will prove the energy estimates for the linear operator (4.6).
Lemma 4.2. Let |�| > 0, ` 2 R, 0 # 2 and 0 < q. If # = 2 restrict 0 < q < 1.
Then 8⌘ > 0 9C(⌘) > 0 such that
hw2(`,#)@�[⌫g], @�gi � |@�g|2⌫,`,# � ⌘X
|�1||�||@�1g|2⌫,`,# � C(⌘)
�
��C(⌘)
g�
�
2
`.
Furthermore, for any |�| � 0 we have
|hw2(`,#)@�[Kg1
], g2
i|
8
<
:
⌘X
|�1||�||@�1g1
|⌫,`,# + C(⌘)�
��C(⌘)
g1
�
�
`
9
=
;
|g2
|⌫,`,#,
where we are using (4.19).
Some parts of the proof of Lemma 4.2 are exactly the same as in [37]. For instance,
the proof of the lower bound for hw2(`,#)@�[⌫g], @�gi is exactly the same. But the
estimate for |hw2(`,#)@�[Kg1
], g2
i| requires extra care in particular because g1
in the
argument of [Kg1
] does not depend only on v. We therefore need to control the new
exponentially growing factor of w(`,#)(v). This requires a close look at the variables
from (4.20) in (4.21). In particular, we write down the analogue of the conservation
of energy (4.4) in this new coordinate system (4.29) in order absorb the exponentially
growing weight. For completeness, we present all details of the proof.
107
Proof. The First Estimate in the Lemma.
Fix ⌘ > 0. Recall
hw2@�[⌫g], @�gi = hw2⌫@�g, @�gi+X
0<�1�
C�1� hw2@�1⌫@���1g, @�gi.
By Lemma 2 in [38], for |�1
| > 0,
|@�1⌫| C(1 + |v|2) ��12 .
We use this estimate (for m is chosen large enough) to obtain
hw2@�1⌫@���1g, @�gi =
Z
|v|m
+
Z
|v|�m
Z
|v|m
+C
m|@���1g|⌫,`,# |@�g|⌫,`,#
Z
|v|m
+⌘
2|@���1g|⌫,`,# |@�g|⌫,`,# .
On the other hand, for such a m > 0 and � � �1
< �, the first integral over |v| m
is bounded by a compact Sobolev interpolation
(4.24)
Z
|v|m
⌘
2
X
|�1|=|�||@�1g|2⌫,` + C(⌘)
�
��C(⌘)
g�
�
2
⌫,`.
This concludes the lower bound for hw2(`,#)@�[⌫g], @�gi.
The Second Estimate in the Lemma. The proof of the second estimate is divided
into several parts. Recall K = K1
�K2
.
Step 1. The Estimate for K1
.
Next consider K1
from (4.17). We change variables u ! u + v to obtain
[K1
g1
](v) =
Z
R3⇥S2
B(✓)|u|�µ1/2(u + v)µ1/2(v)g1
(u + v)dud!
Notice that now cos ✓ = u · !/|u| so that for |�| > 0, @�[K1
g1
](v)
=X
�1�
C�1�
Z
R3⇥S2
B(✓)|u|�@���1
�
µ1/2(u + v)µ1/2(v)�
@�1g1
(u + v)dud!.
For any 0 < q0 < 1 we have
|@���1{µ1/2(u + v)µ1/2(v)}| C(|�|, q0)µq0/2(u + v)µq0/2(v).
108
We will use this exponential decay to control half of the exponential growth in the
weight. If 0 # < 2 then
w(`,#)(v)µq0/2(v) Cµq0/4(v).
If # = 2 then for given 0 < q < 1 we choose q0 so that q < q0 < 1. And in this case
w(`,#)(v)µq0/2(v) = (1 + |v|2)⌧`/2eq
4 eq
4 |v|2µq0/2(v) Cµ(q0�q)/4(v).
Choosing 0 < q00 < min{|q0 � q|/4, q0/4}, we can always write hw2(`,#)@�[K1
g1
], g2
i
=X
�1�
Z
R3⇥R3
w(`,#)(v)|u|�µq00(u + v)µq00(v)µ�1(u + v, v)@�1g1
(u + v)g2
(v)dudv,
where µ�1(u + v, v) is a collection of smooth functions satisfying
�
�
�
@u¯�µ�1(u + v, v)
�
�
�
C(|�|, q, q0, q00).
Change variables u ! u� v to obtain
X
�1�
Z
R3⇥R3
w(`,#)(v)|u� v|�µq00(u)µq00(v)µ�1(u, v)@�1g1
(u)g2
(v)dudv,
Now further split
hw2(`,#)@�[K1
g1
], g2
i = hw2(`,#)@�[K�1
g1
], g2
i+ hw2(`,#)@�[K1��1
g1
], g2
i
= K�1
+ K1��1
.
Using (4.19) we have
K1��1
⌘Z
w(`,#)(v)�m(|u� v|)|u� v|�µq00(u)µq00(v)µ�1(u, v)@�1g1
(u)g2
(v)dudv,
where �m = 1� �m and we implicitly sum over �1
�. Then
�
�K1��1
�
� C
⇢
Z
w2(`,#)(v)�m(|u� v|)|u� v|�µq00(u)µq00(v) |g2
(v)|2 dudv
�
1/2
⇥X
�1�
⇢
Z
�m(|u� v|)|u� v|�µq00(u)µq00(v) |@�1g1
(u)|2 dudv
�
1/2
.
C(2m)3+�
2 |g2
|⌫,`,#
X
�1�
(2m)3+�
2 |@�1g1
|⌫,`
⌘
2|g
2
|⌫,`,#
X
�1�
|@�1g1
|⌫,`.
109
The last step follows by choosing m small enough.
Further,
K�1
⌘Z
w(`,#)(v)�m(|u� v|)|u� v|�µq00(u)µq00(v)µ�1(u, v)@�1g1
(u)g2
(v)dudv,
where we again implicitly sum over �1
�. After an integration by parts
K�1
=X
�1�
(�1)|�1|Z
w(`,#)(v)@u�1
n
�m(|u� v|)|u� v|�µq00(u)µ�1(u, v)o
⇥µq00(v)g1
(u)g2
(v)dudv.
Since |u� v|� is bounded now, from (4.19), choosing m0 > 0 large enough we have
|K�1
| C(|�|, m)
Z
w(`,#)(v)µq00/2(u)µq00/2(v) |g1
(u)g2
(v)| dudv
=
Z
|u|m0+
Z
|u|>m0
C
Z
w(`,#)(v)�m0(|u|)µq00/2(u)µq00/2(v) |g1
(u)g2
(v)| dudv
+ Ce�q
008 m0
Z
w(`,#)(v)µq00/4(u)µq00/2(v) |g1
(u)g2
(v)| dudv
n⌘
2|g
1
|⌫,` + C(m0)|�m0g1
|⌫,`
o
|g2
|⌫,`,#.
This completes the estimate for hw2(`,#)@�[K1
g1
], g2
i and step one.
Step 2. The Estimate for K2
.
We turn to K2
from (4.18). Split K2
= K�2
+ K1��2
and consider K�2
in (4.20).
Step (2a). The Estimate of K1��2
.
Now consider K1��2
= K2
�K�2
which is given by
K1��2
g1
⌘Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)�m(|u� v|)µ1/2(u0)g1
(v0)dud!
+
Z
R3⇥S2
B(✓)|u� v|�µ1/2(u)�m(|u� v|)µ1/2(v0)g1
(u0)dud!.
Here �m = 1� �m and �m is defined in (4.19). (4.3) and {|u� v| 2m} imply
|u0| = |v + u� v � [(u� v) · !]!| � |v|� 2|u� v| � |v|� 4m,
|v0| = |v + [(u� v) · !]!| � |v|� |u� v| � |v|� 2m.
110
Therefore for any 0 < q0 < 1 we have
(4.25) µ1/2(u)µ1/2(u0) + µ1/2(u)µ1/2(v0) eC(q0)m2µ1/2(u)µq0/2(v).
This will be the key point in estimating the K1��2
part.
First we take look at @�[K1��2
g1
]. Change variables u� v ! u to obtain
K1��2
g1
⌘Z
R3⇥S2
B(✓)|u|�µ1/2(u + v)�m(|u|)µ1/2(v + u?)g1
(v + uk)dud!
+
Z
R3⇥S2
B(✓)|u|�µ1/2(u + v)�m(|u|)µ1/2(v + uk)g1
(v + u?)dud!,
Note that uk and u? are defined using notation from [37]:
(4.26) uk ⌘ [u · !]!, u? ⌘ u� [u · !]!.
Now derivatives will not hit the singular kernel. @�[K1��2
g1
] is
C�1�
Z
R3⇥S2
B(✓)|u|��m(|u|)@���1{µ1/2(u + v)µ1/2(v + u?)}@�1g1
(v + uk)dud!
+C�1�
Z
R3⇥S2
B(✓)|u|��m(|u|)@���1{µ1/2(u + v)µ1/2(v + uk)}@�1g1
(v + u?)dud!,
where we implicitly sum over multi-indices �1
�. Therefore, for any 0 < q00 < 1,
|@�[K1��2
g1
]| is bounded by
C
Z
R3⇥S2
|u|��m(|u|)µq00/2(u + v)µq00/2(v + u?)|@�1g1
(v + uk)|dud!
+C
Z
R3⇥S2
|u|��m(|u|)µq00/2(u + v)µq00/2(v + uk)|@�1g1
(v + u?)|dud!.
We change variables u ! u� v again to see that |@�[K1��2
g1
]| is bounded by
C
Z
R3⇥S2
|u� v|��m(|u� v|)µq00/2(u)µq00/2(v0)|@�1g1
(u0)|dud!
+C
Z
R3⇥S2
|u� v|��m(|u� v|)µq00/2(u)µq00/2(u0)|@�1g1
(v0)|dud!.
Use (4.25) with 0 < q0 < q00 to say this is bounded above by
C
Z
R3⇥S2
|u� v|��m(|u� v|)µq00/2(u)µq0/2(v) {|@�1g1
(u0)|+ |@�1g1
(v0)|} dud!.
111
We remark that this last bound is true (and trivial) when |�| = 0 in which case
@�1 = @0
= 1 by convention. Thus, |hw2(`,#)@�{K1��2
g1
}, g2
i| is
C
Z
R3⇥R3⇥S2
|u� v|��m(|u� v|)w2(`,#)(v)µq00/2(u)µq0/2(v)|@�1g1
(v0)||g2
(v)|,
+C
Z
R3⇥R3⇥S2
|u� v|��m(|u� v|)w2(`,#)(v)µq00/2(u)µq0/2(v)|@�1g1
(u0)||g2
(v)|.
Here again we need to control the large exponentially growing factor w(`,#)(v) by
the strong exponential decay of the Maxwellians.
We will control this growth in cases. If 0 # < 2 then q0 > 0 means
w(`,#)(v)µq0/2(v) Cµq0/4(v).
Alternatively, if # = 2 with 0 < q < 1 then we can choose q0 and q00 such that
q < q0 < q00. Then
w(`,#)(v)µq0/2(v) Cw(`, 0)(v)µ(q0�q)/2(v) Cµ(q0�q)/4(v).
In either case, choose q1
= min{q00/2, q0/4, |q0 � q|/4} > 0. Then we have the upper
bound of
C
Z
|u� v|��m(|u� v|)w(`,#)(v)µq1(u)µq1(v)|@�1g1
(v0)||g2
(v)|dvdud!
+C
Z
|u� v|��m(|u� v|)w(`,#)(v)µq1(u)µq1(v)|@�1g1
(u0)||g2
(v)|dvdud!.
Further note that
Z
|u� v|��m(|u� v|)µq1(u)du Cm3+�.
Apply Cauchy-Schwartz and the last display to obtain the upper bound
Cm3+�
2
⇢
Z
|u� v|��m(|u� v|)µq1(u)µq1(v)|@�1g1
(v0)|2dvdud!
�
1/2
|g2
|⌫,`,#
+Cm3+�
2
⇢
Z
|u� v|��m(|u� v|)µq1(u)µq1(v)|@�1g1
(u0)|2dvdud!
�
1/2
|g2
|⌫,`,#.
112
Now apply the change of variables (u, v) ! (u0, v0) using |u� v| = |u0 � v0| and (4.4)
to see that |hw2(`,#)@�{K1��2
g1
}, g2
i| is bounded by
Cm3+�
⇢
Z
|u� v|��m(|u� v|)µq1(u)µq1(v)|@�1g1
(v)|2dvdu
�
1/2
|g2
|⌫,`,#.
Hence,
|hw2(`,#)@�{K1��2
g1
}, g2
i| Cm�+3|g2
|⌫,`,#
X
|�1||�||@�1g1
|⌫,`.
For m > 0 small enough, we have completed the estimate of K1��2
, step (2a).
Step (2b). Estimate of K�2
.
For some large but fixed m0 > 0 we define the smooth cuto↵ function
⌥m0 = ⌥m0(v, ⇠) = �m0(p
1 + |v|2 + |v + ⇠|2), ⌥m0 = 1�⌥m0(v, ⇠).
Now split again K�2
= K⌥
2
+ K1�⌥
2
where
K⌥
2
g1
=
Z
R3
⌥m0(v, ⇠)k�2
(v, ⇠)g1
(v + ⇠)d⇠.
We will estimate this term first. Taking derivatives
@�[K⌥
2
g1
] =X
�1�
C�1�
Z
R3
@v�1
[⌥m0(v, ⇠)k�2
(v, ⇠)]@���1g1
(v + ⇠)d⇠
Using Lemma 4.1, with 0 < s1
< s2
< 1, |hw2(`,#)@�[K⌥
2
g1
], g2
i| is bounded by
(4.27) CX
�1�
Z
|v|+|v+⇠|>m0
w2(`,#)(v)|@���1g1
(v + ⇠)||g2
(v)||⇠|(1 + |v|+ |v + ⇠|)1��
e�s28 |⇠|2� s1
2 |⇣|||2d⇠dv.
By (4.10) we expand
(4.28) w(`,#)(v)e�s28 |⇠|2� s1
2 |⇣|||2 = (1 + |v|2)⌧`/2eq
4 (1+|v|2)
#/2e�
s28 |⇠|2� s1
2 |⇣|||2 .
If we can control (4.28) by w(`,#)(v + ⇠) times decay in other directions then we can
estimate (4.27). To do this, we look for an analogue of (4.4) in the variables (4.21).
Using (4.21) we have
(4.29) |v|2 + |v + ⇠ � ⇣?|2 = |v + ⇠|2 + |v � ⇣?|2 = |v + ⇠|2 +
✓
v · ⇠|⇠|
◆
2
.
113
Since 0 # 2 we have
|v|#
|v + ⇠|2 +
✓
v · ⇠|⇠|
◆
2
!#/2
|v + ⇠|# +
�
�
�
�
v · ⇠|⇠|
�
�
�
�
#
.
Thus,
(4.30) eq
4 (1+|v|2)
#/2 eq
4 eq
4 |v|# eq
4 eq
4 |v+⇠|#eq
4 |v· ⇠
|⇠| |# .
Further, from (4.21) notice that
|⇣k|2 =
✓✓
v · ⇠|⇠|
◆
+1
2|⇠|◆
2
� 1
2
✓
v · ⇠|⇠|
◆
2
� 1
4|⇠|2
Therefore with 0 < s1
< s2
we have
e�s28 |⇠|2� s1
2 |⇣|||2 e�s2�s1
8 |⇠|2� s14 (v· ⇠
|⇠|)2
Combine the above with (4.30), to obtain
eq
4 (1+|v|2)
#/2e�
s28 |⇠|2� s1
2 |⇣|||2 eq
4 eq
4 |v+⇠|#eq
4 |v· ⇠
|⇠| |#e�s28 |⇠|2� s1
2 |⇣|||2
Ceq
4 |v+⇠|#eq
4 |v· ⇠
|⇠| |#e�s2�s1
8 |⇠|2� s14 (v· ⇠
|⇠|)2
= Ceq
4 |v+⇠|#e�s2�s1
8 |⇠|2eq
4 |v· ⇠
|⇠| |#� s14 (v· ⇠
|⇠|)2
.
If 0 # < 2 then
eq
4 |v· ⇠
|⇠| |#� s14 (v· ⇠
|⇠|)2
Ce�s18 (v· ⇠
|⇠|)2
.
And if # = 2 then 0 < q < 1 and we can choose s1
with 1 > s1
> q so that
eq
4 |v· ⇠
|⇠| |2� s14 (v· ⇠
|⇠|)2
Ce�s1�q
4 (v· ⇠
|⇠|)2
.
In either case, choosing s3
= min{|s1
� q|, s1
/2} > 0 and plugging these estimates
into (4.28), we conclude that
w(`,#)(v)e�s28 |⇠|2� s1
2 |⇣|||2
C(1 + |v|2)⌧`/2eq
4 (1+|v+⇠|2)
#/2e�
s2�s18 |⇠|2� s3
4 (v· ⇠
|⇠|)2
.(4.31)
114
Next, we estimate (1 + |v|2)⌧`/2 with ⌧ < 0 and ` 2 R. If `⌧ > 0 then (4.29) yields
(1 + |v|2)⌧`/2
1 + |v + ⇠|2 +
✓
v · ⇠|⇠|
◆
2
!⌧`/2
C(1 + |v + ⇠|2)⌧`/2
1 +
✓
v · ⇠|⇠|
◆
2
!⌧`/2
.
Conversely if `⌧ 0 then we split the region into
{|v + ⇠| > 2|v|} [ {|v + ⇠| 2|v|}.
On {|v + ⇠| 2|v|} and `⌧ 0 then
(1 + |v|2)⌧`/2 C(1 + |v + ⇠|2)⌧`/2.
Alternatively, if {|v + ⇠| > 2|v|} then
|⇠| � |v + ⇠|� |v| > |v + ⇠|/2.
We therefore always have
(1 + |v|2)⌧`/2e�s2�s1
8 |⇠|2 e�s2�s1
8 |⇠|2 e�s2�s1
16 |⇠|2e�s2�s1
64 |v+⇠|2 .
In either of these last few cases, since s2
> s1
> q, from (4.31) we have
w(`,#)(v)e�s28 |⇠|2� s1
2 |⇣|||2 C(1 + |v|2)⌧`/2eq
4 (1+|v+⇠|2)
#/2e�
s2�s18 |⇠|2� s3
4 (v· ⇠
|⇠|)2
C(1 + |v + ⇠|2)⌧`/2eq
4 (1+|v+⇠|2)
#/2e�
s2�s116 |⇠|2� s3
8 (v· ⇠
|⇠|)2
= Cw(`,#)(v + ⇠)e�s2�s1
16 |⇠|2� s38 (v· ⇠
|⇠|)2
.
Plug this into (4.27) to obtain the following upper bound for (4.27) of
C
Z
|v|+|v+⇠|>m0
(w(`,#)(v + ⇠)|@���1g1
(v + ⇠)|) (w(`,#)(v)|g2
(v)|)|⇠|(1 + |v|+ |v + ⇠|)1��
e�s2�s1
16 |⇠|2d⇠dv,
115
where we implicitly sum over �1
�. Using Cauchy-Schwartz and translation invari-
ance this is
C
m0
Z
(w(`,#)(v + ⇠)|@���1g1
(v + ⇠)|) (w(`,#)(v)|g2
(v)|)|⇠|(1 + |v|+ |v + ⇠|)��
e�s2�s1
16 |⇠|2d⇠dv
C
m0 |g2
|⌫,`,#
Z
w2(`,#)(v + ⇠)|@���1g1
(v + ⇠)|2|⇠|(1 + |v + ⇠|)��
e�s2�s1
16 |⇠|2d⇠dv
C
m0X
�1�
|@���1g1
|⌫,`,#|g2
|⌫,`,#.
This completes the estimate for K⌥
2
.
We now estimate K1�⌥
2
. Taking derivatives
@�[K1�⌥
2
g1
] =X
�1�
C�1�
Z
R3
@v�1
[⌥m0(v, ⇠)k�2
(v, ⇠)]@���1g1
(v + ⇠)d⇠
Also,
hw2@�[K1�⌥
2
g1
], g2
i =
Z
w2(`,#)@�[K1�⌥
2
g1
]g2
(v)dv.
By Cauchy-Schwartz and the compact support of K1�⌥
2
we have
�
�hw2@�[K1�⌥
2
g1
], g2
i�
� C(m0)⇢
Z
|v|m0
�
@�[K1�⌥
2
g1
]�
2
dv
�
1/2
|g2
|⌫,`.
With Lemma 4.1 we establish that @�[K1�⌥
2
g1
] is compact from Hk to Hk. Then by
the general interpolation for compact operators from Hk to Hk we have
�
�hw2@�[K1�⌥
2
g1
], g2
i�
�
8
<
:
⌘
4
X
|�1|=|�||@�1g1
|⌫,` + C(⌘, m0) |g1
|⌫,`
9
=
;
|g2
|⌫,`.
This completes the estimate for K1�⌥
2
and thus for K�2
, step (2b). We have therefore
finished the whole proof. ⇤
The following Corollary is used to prove existence of global solutions.
Corollary 4.1. Let |�| > 0, ` 2 R, 0 # 2 and 0 < q. If # = 2 restrict
0 < q < 1. Then 8⌘ > 0 there exists C(⌘) > 0 such that
hw2@�[Lg], @�gi � |@�g|2⌫,`,# � ⌘X
|�1||�||@�1g|2⌫,`,# � C(⌘)
�
��C(⌘)
g�
�
2
`,
where �C(⌘)
is from (4.19).
116
The rest of section is devoted to estimates for nonlinear collision term �[g1,g2
],
with gi(x, v) (i = 1, 2). In (4.5), the (non-symmetric) bilinear form �[g1,g2
] in the
Boltzmann case is
�[g1,g2
] = µ�1/2(v)Q[µ1/2g1
, µ1/2g2
] ⌘ �gain
[g1,g2
]� �loss
[g1,g2
],
=
Z
R3
|u� v|�µ1/2(u)
Z
S2
B(✓)g1
(u0)g2
(v0)d!�
du,(4.32)
�g2
(v)
Z
R3
|u� v|�µ1/2(u)
Z
S2
B(✓)d!
�
g1
(u)du.
The change of variables u� v ! u gives
@↵��[g
1,g2
] ⌘ @↵�
Z
R3
Z
S2
|u|�µ1/2(u + v)g1
(v + uk)g2
(v + u?)B(✓)dud!
�
,
�@↵�
Z
R3
Z
S2
|u|�µ1/2(u + v)g1
(v + u)g2
(v)B(✓)dud!
�
,
⌘X
C�0�1�2� C↵1↵2
↵ �0[@↵1�1
g1,@
↵2�2
g2
].
where the summation is over �0
+�1
+�2
= � and ↵1
+↵2
= ↵. Also u?, uk are given
by (4.26). By the product rule and the reverse change of variables we have
�0[@↵1�1
g1,@
↵2�2
g2
], ⌘Z
R3
Z
S2
|u� v|�@�0 [µ1/2(u)]@↵1
�1g1
(u0)@↵2�2
g2
(v0)B(✓)
�@↵2�2
g2
(v)
Z
R3
Z
S2
|u� v|�@�0 [µ1/2(u)]@↵1
�1g1
(u)B(✓)(4.33)
⌘ �0
gain
� �0
loss
.
With these formulas, we have the following nonlinear estimate:
Lemma 4.3. Recall (4.33) and let �0
+ �1
+ �2
= �, ↵1
+ ↵2
= ↵. Say 0 # 2,
0 < q. If # = 2, restrict 0 < q < 1. Let ` = |�|� l with l � 0. If |↵1
| + |�1
| N/2,
then
|(w2(`,#)�0[@↵1�1
g1,@
↵2�2
g2
], @↵� g
3
)| CE1/2
l,# (g1
)||@↵2�2
g2
||⌫,|�2|�l,#||@↵� g
3
||⌫,`,#.
Alternatively, if |↵2
|+ |�2
| N/2, then
|(w2(`,#)�0[@↵1�1
g1,@
↵2�2
g2
], @↵� g
3
)| C||@↵1�1
g1
||⌫,|�1|�l,#E1/2
l,# (g2
)||@↵� g
3
||⌫,`,#.
117
The proof of Lemma 4.3 is more or less the same as in [37]. However, small
modifications are needed to facilitate the exponentially growing weight. In (4.37) we
need to use (4.4) to properly distribute the exponentially growing factor in w2(`,#)(v).
Proof. Case 1. The Loss Term Estimate.
First consider the second term �0
loss
in (4.33). Note that
|@�0 [µ1/2(u)]| Ce�|u|
2/8.
With |↵1
|+ |�1
| N/2 and � > �3 we haveZ
R3
|u� v|�|@�0 [µ1/2(u)]@↵1
�1g1
(x, u)|du
C
⇢
Z
R3
|u� v|�e�|u|2/8|@↵1�1
g1
(x, u)|2du
�
1/2
⇢
Z
R3
|u� v|�e�|u|2/8du
�
1/2
C supx,u
�
�
�
e�|u|2/16@↵1
�1g1
(x, u)�
�
�
⇢
Z
R3
|u� v|�e�|u|2/16du
�
CE1/2
l (g1
)[1 + |v|]�.
Since N � 8, we have used the embedding H4(T3 ⇥ R3) ⇢ L1 to argue that
(4.34) supx,u
�
�
�
e�|u|2/16@↵1
�1g1
(x, u)�
�
�
CE1/2
l (g1
).
Hence�
�(w2(`,#)�0
loss
[@↵1�1
g1,@
↵2�2
g2
], @↵� g
3
)�
� is bounded by
CE1/2
l (g1
)
Z
[1 + |v|]�w2(`,#)(v)|@↵2�2
g2
(v)@↵� g
3
(v)|dvdx
CE1/2
l (g1
)||@↵2�2
g2
||⌫,`,#||@↵� g
3
||⌫,`,#.
This completes the estimate for �0
loss
when |↵1
|+ |�1
| N/2.
Now consider �0
loss
with |↵2
| + |�2
| N/2. Here we split the (u, v) integration
domain is split into three parts
{|v � u| |v|/2} [ {|v � u| � |v|/2, |v| � 1} [ {|v � u| � |v|/2, |v| 1}.
In the first region, |u| is comparable to |v| and thus we can use exponential decay
in both variables to get the estimate. In the second and third regions, |u| is not
comparable to |v| but we exploit the largeness or smallness of |v| to get the estimate.
118
Case (1a). The Loss Term in the First Region {|v � u| |v|/2}.
For the first part, {|v � u| |v|/2}, we have
|u| � |v|� |v � u| � |v|/2.
So that
e�|u|2/8 e�|u|
2/16e�|v|2/64.
Then the integral of w2�0
loss
[@↵1�1
g1,@
↵2�2
g2
]@↵� g
3
over {|u� v| |v|/2} is bounded by
C
Z
|u� v|�e�|u|2/16e�|v|2/64w2(`,#)(v)|@↵1
�1g1
(u)@↵2�2
g2
(v)@↵� g
3
(v)|dudvdx
C
⇢
Z
|u� v|�e�|u|2/16e�|v|2/64|@↵1
�1g1
(u)|2dudvdx
�
1/2
⇥⇢
Z
|u� v|�e�|u|2/16e�|v|2/64w4(`,#)(v)|@↵2
�2g2
(v)|2|@↵� g
3
(v)|2dudvdx
�
1/2
.
Integrating over dv, the first factor is bounded by C||@↵1�1
g1
||⌫,`. Integrating first over
u variables in the second factor yields an upper bound
C
⇢
Z
✓
Z
|u� v|�e�|u|2/16du
◆
e�|v|2/64w4|@↵2
�2g2
|2|@↵� g
3
|2dvdx
�
1/2
C
⇢
Z Z
e�|v|2/64w4(`,#)(v)[1 + |v|]�|@↵2
�2g2
|2|@↵� g
3
|2dvdx
�
1/2
.
And as in (4.34), since N � 8 and |↵2
|+ |�2
| N/2 we have
(4.35) supx,v
w(`,#)(v)e�|v|2/256|@↵2
�2g2
(x, v)| CE1/2
l,# (g2
).
We thus conclude the estimate over the first region.
Case (1b). The Loss Term in the Second Region {|v � u| � |v|/2, |v| � 1}.
Next consider �0
loss
over the second region {|v� u| � |v|/2, |v| � 1}. Since � < 0,
we have
|u� v|� C[1 + |v|]�.119
Then the integral of w2(`,#)�0
loss
[@↵1�1
g1,@
↵2�2
g2
]@↵� g
3
over this region is bounded by
C
Z
[1 + |v|]�e�|u|2/8w2(`,#)|@↵1�1
g1
(u)@↵2�2
g2
(v)@↵� g
3
(v)|dudvdx
C
Z
⇢
Z
e�|u|2/8|@↵1
�1g1
(u)|du
�⇢
Z
[1 + |v|]�w2(`,#)|@↵2�2
g2
(v)@↵� g
3
(v)|dv
�
dx
C
Z
|@↵1�1
g1
|⌫,`
⇢
Z
w2|@↵2�2
g2
|2dv
�
1/2
⇢
Z
[1 + |v|]2�w2|@↵� g
3
|2dv
�
1/2
dx.
Since |↵2
|+ |�2
| N/2, N � 8 and H2(T3) ⇢ L1(T3), we have
(4.36) supx
Z
w2(`,#)(v)|@↵2�2
g2
(x, v)|2dv CEl,#(g2
).
Thus, by the Cauchy-Schwartz inequality, the �0
loss
term over {|v�u| � |v|/2, |v| � 1}
is bounded by C||@↵1�1
g1
||⌫,`E1/2
l,# (g2
)||@↵� g
3
||⌫,`,#.
Case (1c). The Loss Term in the Third Region: {|v � u| � |v|/2, |v| � 1}
For the last region, {|v � u| � |v|/2, |v| 1}, we have
|u� v|� C|v|�, w(`,#)(v) C.
And then that the integral of w2(`,#)�0
loss
[@↵1�1
g1,@
↵2�2
g2
]@↵� g
3
over this region is bounded
by
C
Z
{|v�u|�|v|/2, |v|1}|u� v|�e�|u|2/8|@↵1
�1g1
(u)@↵2�2
g2
(v)@↵� g
3
(v)|dudvdx
C
Z
⇢
Z
|u� v|�/2e�|u|2/8|@↵1
�1g1
(u)|du
�⇢
Z
|v|1
|v|�/2|@↵2�2
g2
(v)@↵� g
3
(v)|dv
�
dx.
Using the Cauchy-Schwartz inequality a few times we have
C
Z
⇢
Z
|u� v|�e�|u|2/8du
�
1/2
⇢
Z
e�|u|2/8|@↵1
�1g1
(u)|2du
�
1/2
⇥⇢
Z
|v|1
|v|�|@↵2�2
g2
|2dv
�
1/2
⇢
Z
|v|1
|@↵� g
3
|2dv
�
1/2
dx
C
Z
|@↵1�1
g1
|⌫,`
⇢
Z
|v|1
|v|�|@↵2�2
g2
|2dv
�
1/2
⇢
Z
|v|1
|@↵� g
3
|2dv
�
1/2
dx.
By |↵2
|+ |�2
| N/2, � > �3 and H4(T3 ⇥ R3) ⇢ L1,Z
|v|1
|v|�|@↵2�2
g2
|2dv C sup|v|1,x2T3
|@↵2�2
g2
|2 CEl(g2
).
120
Hence, by Cauchy-Schwartz, the last part is bounded by
C||@↵1�1
g1
||⌫,`E1/2
l (g2
)||@↵� g
3
||⌫,`.
This concludes the desired estimate for �0
loss
.
Case 2. The Gain Term Estimate.
The next step is to estimate the gain term �0
gain
in (4.33), for which the (u, v)
integration domain is split into two parts:
{|u| � |v|/2} [ {|u| |v|/2}.
Case (2a) The Gain Term over {|u| � |v|/2}.
For the first region {|u| � |v|/2},
e�|u|2/8 e�|u|
2/16e�|v|2/64.
Then the integral of w2(`,#)�0
gain
[@↵1�1
g1
, @↵2�2
g2
]@↵3�3
g3
over {|u| � |v|/2} is thus bounded
byZ
|u|�|v|/2
|u� v|�e�|u|2/16e�|v|2/64w2(`,#)(v)|@↵1
�1g1
(u0)@↵2�2
g2
(v0)@↵� g
3
(v)|d!dudvdx
C
⇢
Z
|u� v|�e�|u|2/16e�|v|2/64w2|@↵1
�1g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudvdx
�
1/2
⇥⇢
Z
|u� v|�e�|u|2/16e�|v|2/64w2|@↵
� g3
(v)|2dudvdx
�
1/2
C
⇢
Z
|u0 � v0|�e� 164 (|u0|2+|v0|2)w2(v)|@↵1
�1g1
(u0)|2|@↵2�2
g2
(v0)|2�
1/2
||@↵� g
3
||⌫,`,#.
In the first factor we have used (4.4) and |u0 � v0| = |u� v|. By (4.4),
eq
4 (1+|v|2)
#/2 eq
4 (1+|v0|2+|u0|2)
#/2 eq
4 (1+|v0|2)
#/2e
q
4 (1+|u0|2)
#/2.(4.37)
Using this, (4.10) and (4.4) we have
e�164 (|u0|2+|v0|2)w2(`,#)(v) e�
1128 (|u0|2+|v0|2)w2(`,#)(v0)w2(`,#)(u0).
So that the factor in braces is
C
Z
|u0 � v0|�e� 1128 (|u0|2+|v0|2)w2(u0)|@↵1
�1g1
(u0)|2w2(v0)|@↵2�2
g2
(v0)|2d!dudvdx.
121
The change of variables (u, v) ! (u0, v0) implies
= C
⇢
Z
|u� v|�e� 1128 (|u|2+|v|2)w2(u)|@↵1
�1g1
(u)|2w2(v)|@↵2�2
g2
(v)|2dudvdx
�
1/2
.
Assume |↵1
|+ |�1
| N/2. As in (4.35),
supx,u
n
w(`,#)(u)e�1
256 |u|2|@↵1�1
g1
(x, u)|o
CE1/2
l,# (g1
).
Integrate first over du to obtain
Z
|u� v|�e� 1128 |u|2w(`,#)(u)|@↵1
�1g1
(u)|2du CE1/2
l,# (g1
)
Z
|u� v|�e� 1256 |u|2du
CE1/2
l,# (g1
)[1 + |v|]�.
If |↵2
| + |�2
| N/2 use this last argument but switch @↵1�1
g1
with @↵2�2
g2
. Then the
bound for the gain term over {|u| � |v|/2}, if |↵1
|+ |�1
| N/2, is
Z
|u|�|v|/2
CE1/2
l,# (g1
)||@↵2�2
g2
||⌫,`,#||@↵� g
3
||⌫,`,#.
And the bound for the gain term over {|u| � |v|/2}, if |↵2
|+ |�2
| N/2, is
Z
|u|�|v|/2
C||@↵2�2
g1
||⌫,`,#E1/2
l,# (g2
)||@↵� g
3
||⌫,`,#.
This completes the estimate for the gain term over {|u| � |v|/2}.
Case (2b). The Gain Term over {|u| |v|/2}.
Now consider the gain term over {|u| |v|/2}. Since |v � u| < |v|/2 implies
|u| � |v|� |v � u| > |v|/2, we obtain
{|u| |v|/2} = {|u| |v|/2} [ {|v � u| � |v|/2}.122
Further assume |v| 1, then |u| 1/2 and the gain term is bounded by
Z
|v|1,|u||v|/2
|u� v|�e�|u|2/8w2|@↵1�1
g1
(u0)@↵2�2
g2
(v0)@↵� g
3
(v)|d!dudvdx(4.38)
C
Z
|v|1
⇢
|v|�/2
Z
|u|1/2
|u� v|�/2e�|u|2/8|@↵1
�1g1
(u0)@↵2�2
g2
(v0)|d!du
�
|@↵� g
3
(v)|dvdx
C
Z
|v|1
⇢
|v|�Z
|u|1/2
|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!du
�
1/2
⇥⇢
Z
|u|1/2
|u� v|�e�|u|2/4du
�
1/2
|@↵� g
3
(v)|dvdx
C
Z
|v|1
⇢
|v|�Z
|u|1/2
|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!du
�
1/2
|@↵� g
3
(v)|dvdx
C
⇢
Z
|v|1,|u|1/2
|v|�|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudvdx
�
1/2
k@↵� g
3
k⌫,`.
We now estimate the first factor. Since |u| |v|/2, from (4.3) we have
|u0|+ |v0| 2[|u|+ |v|] 3|v|.
Since � < 0, this implies
|v|� 3��|u0|�, |v|� 3��|v0|�.
Thus,
Z
|v|1,|u|1/2
|v|�|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudv
C
Z
|v0|3,|u0|3
min[|v0|�, |u0|�]|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudvdx.
Now change variables (v, u) ! (v0, u0) so that the above is
C
Z
|v|3,|u|3
min[|v|�, |u|�]|@↵1�1
g1
(u)|2|@↵2�2
g2
(v)|2d!dudvdx.
Assume |↵1
|+ |�1
| N/2 and majorize the above by
C
Z
⇢
Z
|u|3
|u|�|@↵1�1
g1
(u)|2du
�⇢
Z
|v|3
|@↵2�2
g2
(v)|2dv
�
dx
C supx,|u|3
|@↵1�1
g1
(u)|2k@↵2�2
g2
k2
⌫,` CEl(g1
)k@↵2�2
g2
k2
⌫,`.
123
Alternatively, if |↵2
|+ |�2
| N/2 then
C
Z
⇢
Z
|u|3
|@↵1�1
g1
(u)|2du
�⇢
Z
|v|3
|v|�|@↵2�2
g2
(v)|2dv
�
dx
Ck@↵1�1
g1
k2
⌫,` supx,|v|3
|@↵2�2
g2
(v)|2 Ck@↵1�1
g1
k2
⌫,`El(g2
).
Combine this upper bound with (4.38) to complete the estimate for the gain term
over {|u| |v|/2, |v| 1}.
Case (2c) The Gain Term over {|u| |v|/2, |v � u| � |v|/2, |v| � 1}.
The last case is the gain term over the region {|u| |v|/2, |v�u| � |v|/2, |v| � 1}.
The integral of w2(`,#)�0
gain
[@↵1�1
g1
, @↵2�2
g2
]@↵� g
3
over such a region is bounded byZ
|u||v|/2,|v|�1
|u� v|�e�|u|2/4w2(`,#)|@↵1�1
g1
(u0)@↵2�2
g2
(v0)@↵� g
3
(v)|d!dudvdx
C
Z
|u||v|/2,|v|�1
[1 + |v|]�e�|u|2/4w2(`,#)|@↵1�1
g1
(u0)@↵2�2
g2
(v0)@↵� g
3
(v)|d!dudvdx
C
⇢
Z
[1 + |v|]�e�|u|2/4w2(`,#)|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudvdx
�
1/2
⇥⇢
Z
[1 + |v|]�e�|u|2/4w2(`,#)|@↵� g
3
(v)|2d!dudvdx
�
1/2
(4.39)
C
⇢
Z
[1 + |v|]�w2(`,#)(v)|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudvdx
�
1/2
||@↵� g
3
||⌫,`,#.
We have used |v � u|� 4��[1 + |v|]� in the first inequality. If `⌧ < 0, use |v| � 2|u|
and (4.4) to establish
(1 + |v|2)`⌧/2 C(1 + |v0|2 + |u0|2)`⌧/2.
Conversely, if `⌧ � 0, just use (4.4) to establish the same inequality. Recall (4.37)
and M(v) = exp�
q4
(1 + |v|2)#/2
�
. Thus,
w2(`,#)(v) C(1 + |v0|2 + |u0|2)`⌧M(v0)M(u0)
Then since ` = |�|� l we have
w2(`,#)(v) C(1 + |v0|2 + |u0|2)�l⌧ (1 + |v0|2 + |u0|2)|�|⌧M(v0)M(u0)
C(1 + |v0|2)�l⌧ (1 + |u0|2)�l⌧ (1 + |v0|2 + |u0|2)|�|⌧M(v0)M(u0).
124
Assume |↵2
| + |�2
| N/2. Using this estimate and the change of variable (v, u) !
(v0, u0) we obtainZ
[1 + |v|]�w2(`,#)(v)|@↵1�1
g1
(u0)|2|@↵2�2
g2
(v0)|2d!dudvdx
C
Z
[1 + |u0|]�w2(|�1
|� l,#)(u0)|@↵1�1
g1
(u0)|2w2(|�2
|� l,#)(v0)|@↵2�2
g2
(v0)|2
= C
Z
[1 + |u|]�w2(|�1
|� l,#)(u)|@↵1�1
g1
(u)|2w2(|�2
|� l,#)(v)|@↵2�2
g2
(v)|2dudvdx
= C
Z
⇢
Z
w2(|�2
|� l,#)(v)|@↵2�2
g2
(v)|2dv
�
⇥⇢
Z
[1 + |u|]�w2(|�1
|� l,#)(u)|@↵1�1
g1
(u)|2du
�
dx.
Using the embedding in (4.36), we see that this is bounded by
CEl,#(g2
)
Z
[1 + |u|]�w2(|�1
|� l,#)(u)|@↵1�1
g1
(u)|2dudx
CEl,#(g2
)||@↵1�1
g1
||2⌫,|�1|�l,#.
Similarly if |↵1
|+ |�1
| N/2 then this is bounded by
C||@↵2�2
g2
||2⌫,|�2|�l,#El,#(g1
).
Combine this with (4.39) to complete the nonlinear estimate. ⇤
This completes the estimates for the Boltzmann case. In Section 4.5 we use these
to establish global existence. Then in Section 4.6 we prove the decay. In the next
section we establish the analogous estimates for the Landau case.
4.4. Landau Estimates
In this section, we will prove the basic estimates used to obtain global existence of
solutions with an exponential weight in the Landau case. In this case, the derivatives
in the Landau operator cause extra di�culties in particular because @iw(`,#) can
grow faster in v than w(`,#). This new feature of the exponential weight (4.10)
forces us to weaken the linear estimate with high order velocity derivatives (Lemma
4.8) from the analogous estimate [38, Lemma 6, p.403]. A new linear estimate with
no extra derivatives (Lemma 4.9) is also necessary because of the exponential weight.
125
It turns out that we need to dig up exact cancellation in order to prove this estimate
in the # = 2 case. We will again point out the new features of each proof after the
statement of each lemma.
For any vector-valued function g(v) = (g1
, g2
, g3
), we define the projection to the
vector v as
(4.40) Pvgi ⌘vi
|v|
3
X
j=1
vj
|v|gj.
Furthermore, in this section we will use the Einstein summation convention over i
and j, e.g. repeated indices are always summed:
�i(v) = �ij(v)vj
2=
3
X
j=1
�ij(v)vj
2.
With this notation we have
Lemma 4.4. [16, 38] �ij(v), �i(v) are smooth functions such that
|@��ij(v)|+ |@��
i(v)| C�[1 + |v|]�+2�|�|,
and furthermore
(4.41) �ij(v) = �1
(v)vivj
|v|2 + �2
(v)
✓
�ij �vivj
|v|2◆
.
Thus
�ij(v)gigj = �1
(v)3
X
i=1
{Pvgi}2 + �2
(v)3
X
i=1
{[I � Pv]gi}2.(4.42)
Moreover, there are constants c1
and c2
> 0 such that as |v|!1
�1
(v) ⇠ c1
[1 + |v|]�, �2
(v) ⇠ c2
[1 + |v|]�+2.
The estimate of �ij and �i with high derivatives was already established in [16].
The computation of the eigenvalues and their convergence rate was already shown in
[38]. We prove the representation (4.41) below because we will use it in important
places in later proofs and it is not formally written down in the other papers.
126
Proof. Recall from (4.9) that
�ij(v) = (2⇡)�3/2
Z
R3
✓
�ij �(vi � ui)(vj � uj)
|v � u|2◆
|v � u|�+2e�|u|2/2du.
Changing variables u ! v � u we have
�ij(v) = (2⇡)�3/2
Z
R3
✓
�ij �uiuj
|u|2◆
|u|�+2e�|v�u|2/2du.
Given v 2 R3 define v1 = v/|v| and complete an orthonormal basis {v1, v2, v3} where
vi · vj = �ij. Then define the corresponding orthogonal 3⇥ 3 matrix as
O = [v1 v2 v3].
Applying this orthogonal transformation to the integral in �ij above we obtain
�ij(v) = (2⇡)�3/2
Z
R3
✓
�ij �(Ou)i(Ou)j
|u|2◆
|u|�+2e�|v�Ou|2/2du.
Here we have used |Ou| = |u|. Also
(Ou)i = u1
v1
i + u2
v2
i + u3
v3
i .
And
(4.43) |v �Ou|2 = |v � u1
v1 � u2
v2 � u3
v3|2 = (|v|� u1
)2 + u2
2
+ u2
3
.
We therefore have
�ij(v) = (2⇡)�3/2
Z
R3
�ij �3
X
l,m=1
ulum
|u|2 vliv
mj
!
|u|�+2e�12 [(|v|�u1)
2+u2
2+u23]du
= (2⇡)�3/2
Z
R3
�ij �3
X
m=1
u2
m
|u|2vmi vm
j
!
|u|�+2e�12 [(|v|�u1)
2+u2
2+u23]du,
where we have used the odd function argument. Write (m = 1, 2, 3)
B0
(v) ⌘ (2⇡)�3/2
Z
R3
|u|�+2e�12 [(|v|�u1)
2+u2
2+u23]du,
Bm(v) ⌘ (2⇡)�3/2
Z
R3
u2
m
|u|2 |u|�+2e�
12 [(|v|�u1)
2+u2
2+u23]du.
Then by symmetry
B2
(v) = B3
(v) = (2⇡)�3/2
Z
R3
u2
2
+ u2
3
2|u|2 |u|�+2e�12 [(|v|�u1)
2+u2
2+u23]du,
127
and we have
�ij(v) = B0
(v)�ij �B1
(v)v1
j v1
j �B2
(v)(v2
i v2
j + v3
i v3
j ).
Define the orthogonal projections Pj = vj ⌦ vj. Then we have the resolution of the
identity I = P1
+ P2
+ P3
. In component form this is
v2
i v2
j + v3
i v3
j = �ij �vivj
|v|2 .
Then we have
�ij(v) = {B0
(v)�B1
(v)}vivj
|v|2 + {B0
(v)�B2
(v)}✓
�ij �vivj
|v|2◆
.
Now the eigenvalues are �1
(v) = B0
(v)�B1
(v) and �2
(v) = B0
(v)�B2
(v) or
�1
(v) = (2⇡)�3/2
Z
R3
|u|�+2
✓
1� u2
1
|u|2◆
e�12 [(|v|�u1)
2+u2
2+u23]du,
and
�2
(v) = (2⇡)�3/2
Z
R3
|u|�+2
✓
1� u2
2
+ u2
3
2|u|2◆
e�12 [(|v|�u1)
2+u2
2+u23]du.
This completes the derivation of the spectral representation for �ij(v). ⇤
Next, we write bounds for the � norm.
Lemma 4.5. [38, Corollary 1, p.399] There exists c > 0 such that
c|g|2�,`,# ��
�
�
[1 + |v|] �
2 {Pv@ig}�
�
�
2
`,#+�
�
�
[1 + |v|] �+22 {[I � Pv]@ig}
�
�
�
2
`,#+�
�
�
[1 + |v|] �+22 g�
�
�
2
`,#.
Furthermore,
1
c|g|2�,`,#
�
�
�
[1 + |v|] �
2 {Pv@ig}�
�
�
2
`,#+�
�
�
[1 + |v|] �+22 {[I � Pv]@ig}
�
�
�
2
`,#+�
�
�
[1 + |v|] �+22 g�
�
�
2
`,#.
The upper bound was not written down in [38], but the proof is the same. We write
it down here because we will use it in the non-linear estimate. Next, the operators
A, K and � from (4.5) and (4.8) in the Landau case are defined.
128
Lemma 4.6. [38, Lemma 1, p.395] We have the following representations for A, K
and �.
Ag2
= @i[�ij@jg2
]� �ij vi
2
vj
2g2
+ @i�ig
2
(4.44)
Kg1
= �µ�1/2@i{µ[�ij ⇤ {µ1/2[@jg1
+vj
2g1
]}]}(4.45)
= �µ�1/2@i
⇢
µ
Z
R3
�ij(v � v0)µ1/2(v0)[@jg1
(v0) +v0j2
g1
(v0)]dv0�
,
�[g1
, g2
] = @i[{�ij ⇤ [µ1/2g1
]}@jg2
]� {�ij ⇤ [vi
2µ1/2g
1
]}@jg2
�@i[{�ij ⇤ [µ1/2@jg1
]}g2
] + {�ij ⇤ [vi
2µ1/2@jg1
]}g2
.(4.46)
These representations are di↵erent in a few places by a factor of 1
2
from those in
[38]. The only reason for this di↵erence is our use of a di↵erent normalization for the
Maxwellian in this paper.
Proof. We only reprove A. First notice that for either fixed i or j
(4.47)X
i
�ij(v)vi =X
j
�ij(v)vj = 0.
We now take the derivatives inside Ag2
Ag2
= µ�1/2Q(µ, µ1/2g2
)
= µ�1/2@i{�ijµ1/2[@jg2
� vj
2g2
]}+ µ�1/2@i{{�ij ⇤ [vjµ]}µ1/2g2
}
= µ�1/2@i{�ijµ1/2[@jg2
� vj
2g2
]}
+µ�1/2@i{[�ij ⇤ µ]vjµ1/2g
2
} by (4.47)
= µ�1/2@i{�ijµ1/2[@jg2
+vj
2g2
]}
= µ�1/2@i{�ijµ1/2@jg2
}+ µ�1/2@i{�ijµ1/2
vj
2g2
}
= @i[�ij@jg2
] + µ�1/2@i[µ1/2]�ij@jg2
+ @i{�ij vj
2g2
}+ µ�1/2@i[µ1/2]�ij vj
2g2
= @i[�ij@jg2
]� �ij vi
2
vj
2g2
+ µ�1/2@i[µ1/2]�ij@jg2
+ @i{�ij vj
2g2
}
= @i[�ij@jg2
]� �ij vi
2
vj
2g2
+ @i{�ij vj
2}g
2
,
from (4.9), where @i[µ1/2] = vi
2
µ1/2. ⇤129
In the rest of this section, we will prove estimates for A, K and �.
Lemma 4.7. Let 0 # 2, ` 2 R and 0 < q. If # = 2 restrict 0 < q < 1. Then
for any ⌘ > 0, there is 0 < C = C(⌘) < 1 such that
|hw2@i�ig
1
, g2
i|+ |hw2Kg1
, g2
i| ⌘ |g1
|�,`,# |g2
|�,`,# + C|g1
�C |`|g2
�C |`,(4.48)
where w2 = w2(`,#) and �C(⌘)
is defined in (4.19).
The estimate for the @i�i term is exactly the same as in [38]. But, as in the
Boltzmann case, the estimate for K needs modification because g1
in Kg1
does not
depend on v. So we need to show that K can control one exponentially growing factor
w(`,#)(v). We remark that, although it is not used in this paper, the proof clearly
shows |hw2Kg1
, g2
i| ⌘ |g1
|�,` |g2
|�,`,# + C|g1
�C |`|g2
�C |`.
Proof. For m > 0, we split
Z
w2@i�ig
1
g2
=
Z
{|v|m}+
Z
{|v|�m}.
By Lemma 4.4, |@i�i| C[1 + |v|]�+1. Thus, the integral over {|v| m} is
C(m)|g1
�m|`|g2
�m|`. From Lemma 4.5 and the Cauchy-Schwartz inequality
(4.49)
Z
{|v|�m}w2|@i�
ig1
g2
|dv C
m
Z
w2[1 + |v|]�+2|g1
g2
| C
m|g
1
|�,`,# |g2
|�,`,# .
This completes (4.48) for the @i�i term.
Recalling the linear operator K in (4.45), we have
w2Kg1
= �@i{w2µ1/2[�ij ⇤ {µ1/2@jg1
+vj
2µ1/2g
1
}]}
+@i(w2)µ1/2[�ij ⇤ {µ1/2@jg1
+vj
2µ1/2g
1
}](4.50)
+w2
vi
2µ1/2[�ij ⇤ {µ1/2@jg1
+vj
2µ1/2g
1
}].
The derivative of the weight function is
@i(w2(`,#)) = w2(`,#)w
1
(v)vi.(4.51)
130
where
(4.52) w1
(v) =
⇢
2`⌧(1 + |v|2)�1 + q#
2(1 + |v|2)#
2�1
�
.
After integrating by parts for the first term and collecting terms, we can rewrite
hw2Kg1
, g2
i as
X
|�1|,|�2|1
Z
w2(v)�ij(v � v0)µ1/2(v)µ1/2(v0)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv,
where µ�1�2(v, v0) is a collection of smooth functions satisfying
|rvµ�1�2(v, v0)|+ |rv0µ�1�2(v, v0)|+ |µ�1�2(v, v0)| C(1 + |v0|2)1/2(1 + |v|2)1/2.
Since either 0 # < 2 or # = 2 and 0 < q < 1, there exists 0 < q0 < 1 such that
w(`,#)(v)µ1/2(v) Cµq0/2(v)(4.53)
If 0 # < 2 choose any 0 < q0 < 1 and if # = 2 choose 0 < q0 < 1� q. Therefore, we
can rewrite hw2Kg1
, g2
i as
X
|�1|,|�2|1
Z
w(v)�ij(v � v0)µq0/4(v)µ1/4(v0)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv,
where µ�1�2(v, v0) is a di↵erent collection of smooth functions satisfying
|rvµ�1�2(v, v0)|+ |rv0µ�1�2(v, v0)|+ |µ�1�2(v, v0)| Ce�q
016 |v|2e�
116 |v0|2 .
We have removed an exponentially growing factor w(`,#)(v).
Since �ij(v) = O(|v|�+2) 2 L2
loc(R3) and � � �3, Fubini’s Theorem implies
�ij(v � v0)µq0/4(v)µ1/4(v0) 2 L2(R3 ⇥ R3).
Therefore, for any given m > 0, we can choose a C1c function ij(v, v0) such that
||�ij(v � v0)µq0/4(v)µ1/4(v0)� ij(v, v0)||L2(R3⇥R3
)
1
m,
supp{ ij} ⇢ {|v0|+ |v| C(m)} < 1.
We split
�ij(v � v0)µq0/4(v)µ1/4(v0) = ij + [�ij(v � v0)µq0/4(v)µ1/4(v0)� ij].
131
Then
(4.54) hw2Kg1
, g2
i = J1
[g1
, g2
] + J2
[g1
, g2
],
where
J1
=
Z
w(v) ij(v, v0)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv,
J2
=
Z
w(v)[�ij(v � v0)µq0/4(v)µ1/4(v0)� ij]µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv.
Above we are implicitly summing over |�1
|, |�2
| 1. We will bound each of these
terms separately.
The J2
term is bounded as
|J2
| ||�ij(v � v0)µq0/4(v)µ1/4(v0)� ij(v, v0)||L2(R3⇥R3
)
⇥ ||w(v)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)||L2(R3⇥R3
)
C
m
�
�µ1/16@�1g1
�
�
0
�
�
�
µq0/16@�2g2
�
�
�
`,# C
m|g
1
|�,` |g2
|�,`,# .
Now for the first term J1
, integrations by parts over v and v0 variables yields
|J1
| =
�
�
�
�
(�1)�1+�2
Z
@�2 [w(v)@�1{ ij(v, v0)µ�1�2(v, v0)}]g1
(v0)g2
(v)
�
�
�
�
C|| ij||C2
⇢
Z
|v|C(m)
|g1
|2dv
�
1/2
⇢
Z
|v|C(m)
|g2
|2dv
�
1/2
.
We thus conclude (4.48) by choosing m > 0 large enough. ⇤
Next, we estimate the linear terms with velocity derivatives.
Lemma 4.8. Let |�| > 0, ` 2 R, 0 # 2 and q > 0. If # = 2 fix 0 < q < 1.
Then for small ⌘ > 0, there exists C(⌘) > 0 such that
|hw2(`,#)@�[Kg1
], g2
i|
8
<
:
⌘X
|¯�||�|
�
�@¯�g
1
�
�
�,`+ C(⌘)
�
��C(⌘)
g1
�
�
`
9
=
;
|g2
|�,`,# .
Further if ⌧ �1 in (4.10) and ` = r � l where l � 0 and r � |�|, then
�hw2@�[Ag], @�gi � |@�g|2�,`,# � ⌘X
|¯�|=|�|
�
�@¯�g�
�
2
�,`,#� C(⌘)
X
|¯�|<|�|
�
�@¯�g�
�
2
�,|¯�|�l,#,
where w2 = w2(`,#).
132
Notice that the estimate involving [Ag] is much weaker than the analogous esti-
mate [38, Lemma 6, p.403] with no exponential weight. In [38], there are no deriva-
tives in the last term on the right. The key problem here is that derivatives of the
exponential weight, in particular @i(w2(`,#)), can grow faster than w2(`,#). Then,
in some cases, we don’t have enough decay to get the sharper estimate. Instead, we
weaken the estimate and use lower order derivatives to extract polynomial decay from
higher order weights. For the estimate involving [Kg1
] the di↵erence is the same as
in the previous cases; we again show that K controls an exponentially growing factor
of w(`,#)(v). We remark that these estimates are not at all optimal. It is not hard
to see that you can use a smaller norm over a compact region, in particular, for the
terms with no derivatives in the [Ag] estimate.
Proof. We begin with the estimate involving @�[Ag]. Using Lemma 4.6, we have
hw2@�[Ag], @�gi = �|@�g|2�,`,# � C�1� hw2@�1�
ij@���1@jg, @�@igi
�C�2� h@i(w
2)@�2�ij@���2@jg, @�gi(4.55)
�C�1� hw2@�1{�ijvivj}@���1g, @�gi
+C�2� hw2@�2@i�
i@���2g, @�gi.
Here summations are over � � �1
> 0 and � � �2
� 0. We will estimate each of these
terms separately.
Case 1. The Last Two Terms.
First, we consider the last two terms in (4.55). We claim that
|w2@�2@i�i(v)|+ |w2@�1{�ijvivj}| C[1 + |v|]�+1w2,
For the first term on the l.h.s., this follows from Lemma 4.4. For the estimate for the
second term on the r.h.s., from (4.9) and (4.47) we have
�ij(v)vivj =
Z
R3
�ij(v � u)uiujµ(u)du.
133
Now the estimate follows from [38, Lemma 2,p.397] and |�1
| > 0. Using the claim,
the last two terms in (4.55) are bounded by
C
Z
w2[1 + |v|]�+1{|@���1g|+ |@���2g|}|@�g| = C
Z
|v|m
+C
Z
|v|�m
C
Z
|v|m
+C
m
�
�
�
[1 + |v|] �+22 {|@���1g|+ |@���2g|}
�
�
�
`,#
�
�
�
[1 + |v|] �+22 @�g
�
�
�
`,#
C
Z
|v|m
+C
m
X
|¯�||�|
�
�@¯�g�
�
�,`,#|@�g|�,`,# .(4.56)
We have used Lemma 4.5 in the last step. For the part |v| m, for any m0 > 0, we
use the compact Sobolev space interpolation and Lemma 4.5 to get
Z
|v|m
1
m0X
|¯�|=|�|+1
Z
|v|m
|@¯�g|2 + Cm0
Z
|v|m
|g|2
C
m0X
|¯�|=|�|
�
�@¯�g�
�
2
�,`+ Cm0 |�mg|2` .(4.57)
We used Lemma 4.5 again in the last step. This completes the estimate for the last
two terms in (4.55).
Case 2. The Second Term.
Next, we consider the second term in (4.55). Since |�1
| � 1, we have
|hw2@�1�ij@���1@jg, @�@igi| C
Z
[1 + |v|]�+1w2|@���1@jg@�@ig|
C |@�g|�,`,#
⇢
Z
[1 + |v|]�+2w2(`,#)|@���1@jg|2�
1/2
Now using (4.61) with �1
= �2
, given m0 > 0 this is
C |@�g|�,`,#
⇢
Z
[1 + |v|]�w2(|� � �1
|� l,#)|@���1@jg|2�
1/2
(4.58)
C |@�g|�,`,#
X
|¯�||�|�1
�
�@¯�g�
�
�,|¯�|�l,# 1
m0 |@�g|2�,`,# + Cm0
X
|¯�||�|�1
�
�@¯�g�
�
2
�,|¯�|�l,#.
We have now estimated all the terms in (4.55). We conclude case 2 by first choosing
m large enough.
Case 3. The Third Term.
134
Next consider the third and most delicate term in (4.55) when |�2
| = 0. Recall
@i(w2) from (4.51); from (4.52) we have |w1
(v)| C since 0 # 2. And from
(4.41) we have
(4.59) �ij(v)vi = �1
(v)vj
Using Lemma 4.4 for the decay of �1
(v), the third term in (4.55) with |�2
| = 0 is
�
�h@i(w2)�ij@�@jg, @�gi
�
� C
Z
w2(`,#)[1 + |v|]�+1 |@�@jg| |@�g| dv,
= C
Z
⇣
w(`,#)[1 + |v|] �
2 |@�@jg|⌘
�
w(`,#)[1 + |v|](�+2)/2 |@�g|�
dv.
Consider the second term in parenthesis. We will use the weight to extract extra
polynomial decay and look at this as a term with lower order derivatives in the �
norm. Write @� = @��ek
@k were ek is an element of the standard basis. Further, from
(4.10) with ⌧ �1, write out
w(`,#) = (1 + |v|2)⌧`/2 exp⇣q
4(1 + |v|2)#
2
⌘
= w(`� 1,#)(1 + |v|2)⌧/2
Cw(`� 1,#)[1 + |v|]�1.
Since |ek| = 1 and ` = r � l with r � |�|, `� 1 � |�|� 1� l = |� � ek|� l. Thus,
w(`� 1,#)[1 + |v|]�1 w(|� � ek|� l,#)[1 + |v|]�1.
Hence,
w(`,#)[1 + |v|](�+2)/2 w(|� � ek|� l,#)[1 + |v|] �
2 .
Then, for any large m0 > 0, |h@i(w2)�ij@�@jg, @�gi| is
C
Z
⇣
w(`,#)[1 + |v|] �
2 |@�@jg|⌘⇣
w(|� � ek|� l,#)[1 + |v|] �
2 |@��ek
@kg|⌘
dv
C|@�g|�,`,#|@��ek
g|�,|��ek
|�l,#
1
m0 |@�g|2�,`,# + Cm0
X
|¯�|<|�||@
¯�g|2�,|¯�|�l,#.(4.60)
This completes the estimate for the third term in (4.55) when |�2
| = 0.
135
Next consider the third term in (4.55) when |�2
| > 0. Since |�2
| � 1,
�
�@i(w2(`,#))@�2�
ij�
� Cw2(`,#)[1 + |v|]�+2
Notice that the order of @���2@j in this case is < |�|. Again we exploit the lower
order derivative to gain some decay from the weight. Since ⌧ �1, we split
w(`,#) = w(`� 1 + 1,#) = w(`� 1,#)(1 + |v|2)⌧/2
w(|� � �2
|� l,#)[1 + |v|]�1.(4.61)
In this last step we have used ` = r� l, r � |�| so that r�1 � |���2
| since |�2
| � 1.
Given m0 > 0, in this case, the third term in (4.55) has the upper bound
C
Z
�
�@i(w2(`,#))@�2�
ij�
� |@���2@jg@�g| C
Z
w2[1 + |v|]�+2 |@���2@jg@�g|
C |@�g|�,`,#
⇢
Z
w2(`,#)[1 + |v|]�+2 |@���2@jg|2 dv
�
1/2
(4.62)
C |@�g|�,`,#
⇢
Z
w2(|� � �2
|� l,#)[1 + |v|]� |@���2@jg|2 dv
�
1/2
C |@�g|�,`,#
X
|¯�||�|�1
�
�@¯�g�
�
�,|¯�|�l,# 1
m0 |@�g|2�,`,# + Cm0
X
|¯�||�|�1
�
�@¯�g�
�
2
�,|¯�|�l,#.
This completes the estimate for the third term in (4.55). By combining (4.56), (4.57),
(4.60), (4.62) and (4.58) with m and m0 chosen large enough we complete this esti-
mate.
We now estimate hw2@�[Kg1
], g2
i. Recalling (4.45), we have
w2@�Kg1
= �@i[w2@�{µ1/2[�ij ⇤ {µ1/2@jg1
+vj
2µ1/2g
1
}]}]
+@i(w2)@�{µ1/2[�ij ⇤ {µ1/2@jg1
+vj
2µ1/2g
1
}]}
+w2@�{vi
2µ1/2[�ij ⇤ {µ1/2@jg1
+vj
2µ1/2g
1
}]}.
We take derivatives only on the factor {µ1/2@jg1
+vjµ1/2g1
} in the convolutions above.
Upon integrating by parts for the first term, using (4.51) and (4.52) and collecting
136
terms we can express hw2@�[Kg1
], g2
i as
X
|�1||�|+1,|�2|1
Z
w2�ij(v � v0)µ1/2(v)µ1/2(v0)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv,
where µ�1�2(v, v0) is a collection of smooth functions which, for any k�th order deriva-
tives, satisfies
|rkv,v0µ
�1�2(v0, v)| C(1 + |v|2)|�|/2(1 + |v0|2)|�|/2.
Using the same argument as in (4.53) for the same 0 < q0 < 1 as in (4.53) we can
rewrite hw2@�[Kg1
], g2
i as
X
|�1||�|+1,|�2|1
Z
w(v)�ij(v � v0)µq0/4(v)µ1/4(v0)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv,
where µ�1�2(v, v0) is a collection of smooth functions satisfying (for any k�th order
derivatives)
|rkv,v0µ
�1�2(v0, v)| Ce�q
016 |v|2e�
116 |v0|2 .
We split hw2@�[Kg1
], g2
i as in (4.54) to get
X
Z
w(v) ij(v, v0)µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv
+X
Z
w(v){�ij(v � v0)µq0/4(v)µ1/4(v0)� ij}µ�1�2(v, v0)@�1g1
(v0)@�2g2
(v)dv0dv.
Using the estimates as for J2
in (4.54) and Lemma 4.5 the last term is bounded by
C
m
X
|¯�||�|
�
�@¯�g
1
�
�
�,`|g
2
|�,`,# .
Since ij has compact support, integrating by parts over v0 and v, the first term is
equal to
X
|�1||�|+1,|�2|1
(�1)|�1|+|�2|Z
@�2{w(v)@�1 [ ij(v, v0)µ�1�2(v, v0)]}g
1
(v0)g2
(v)dv0dv.
Then, by Cauchy-Schwartz, this term is C(m)�
��C(m)
µg1
�
�
`
�
��C(m)
g2
�
�
`. And our
lemma follows by first choosing m > 0 large. ⇤
Next, from Lemma 4.8 we get a general lower bound for L with high derivatives.
We also prove a lower bound for L with no derivatives.
137
Lemma 4.9. Let 0 # 2, q > 0 and l � 0 with |�| > 0 and ` = |�|� l. If # = 2
further restrict 0 < q < 1. Then for ⌘ > 0 small enough there exists C(⌘) > 0 such
that
hw2@�[Lg], @�gi � |@�g|2�,`,# � ⌘X
|�1|=|�||@�1g|2�,`,# � C(⌘)
X
|�1|<|�||@�1g|2�,|�1|�l,# ,
where w2 = w2(`,#). If |�| = 0 we have
hw2(`,#)[Lg], gi � �q |g|2�,`,# � C(⌘)�
��C(⌘)
g�
�
2
`,
where �q = 1� q2 � ⌘ > 0 for ⌘ > 0 small enough.
It turns out that the lower bound for L with no extra v derivatives and an expo-
nential weight needs a new approach. We need to use exact cancellation to make it
work in the # = 2 case.
Proof. By Lemma 4.8, we need only consider the case with |�| = 0.
First assume 0 # < 2. In this case, after an integration by parts, (4.44) gives
hw2Lg, gi = |g|2�,`,# + h@i(w2)�ij@jg, gi � hw2@i�
ig, gi � hw2Kg, gi.
By Lemma 4.7, the last two terms on the r.h.s. satisfy the |�| = 0 estimate. Thus,
we only consider h@i(w2)�ij@jg, gi. By (4.51) and (4.59) we can write
@i(w2(v))�ij(v) = w2(v)w
1
(v)�1
(v)vj.
From (4.52), |w1
(v)| C(1 + |v|2)#
2�1, and by Lemma 4.4, |�1
(v)vj| C[1 + |v|]�+1.
Thus for any m0 > 0
�
�h@i(w2)�ij@jg, gi
�
� C
Z
w2(`,#)[1 + |v|]�+1+
#
2�1 |@jg| |g| dv
= C
Z
⇣
w(`,#)[1 + |v|] �
2 |@jg|⌘⇣
w(`,#)[1 + |v|] �+22 +
#
2�1 |g|⌘
C |g|�,`,#
�
�
�
[1 + |v|] �+22 +
#
2�1g�
�
�
`,#
1
m0 |g|2
�,`,# + C(m0)�
�
�
[1 + |v|] �+22 +
#
2�1g�
�
�
2
`,#.
138
For m > 0 further split�
�
�
[1 + |v|](�+2)/2+
#
2�1g�
�
�
2
`,#=
Z
|v|m
+
Z
|v|>m
Z
|v|m
+Cm#�2
Z
|v|>m
w2[1 + |v|]�+2|g|2dv
C(m)
Z
|v|m
w2(`, 0)|g|2dv + Cm#�2 |g|2�,`,# .
(4.63)
C(m)|�mg|` + Cm#�2 |g|2�,`,# .
We thus complete the estimate for 0 # < 2 by choosing m and m0 large.
Finally consider the case # = 2 and 0 < q < 1. We will prove this case in two
steps. Split L = �A�K. Define M(v) ⌘ exp�
q4
(1 + |v|2)�
. First we will show that
there is �q > 0 such that
(4.64) �hw2(`, 2)[Ag], gi � �q |Mg|2�,`, � C(�q)�
��C(�q
)
g�
�
2
`.
Second we will establish
(4.65) |Mg|2�,` � �q|g|2�,`,2 � C(�q)|g�C(�q
)
|2` ,
where �q = 1 � q2 � ⌘2
> 0 since ⌘ > 0 can be chosen arbitrarily small. This will be
enough to establish the case # = 2 because the K part is controlled by Lemma 4.7.
We now establish (4.64). By (4.44), we obtain
�M [Ag] = �M@i{�ij@jg}+ M�ij vi
2
vj
2g �M@i�
ig
= �@i{M�ij@jg}+ q�ij vi
2M@jg + M�ij vi
2
vj
2g �M@i�
ig
= �@i{�ij@j[Mg]}+ q@i{�ij vj
2Mg}+ q�ij vi
2M@jg
+ M�ij vi
2
vj
2g �M@i�
ig
= �@i{�ij@j[Mg]}+ q@i{�iMg}+ q�j@j[Mg]
� q2M�ij vi
2
vj
2g + M�ij vi
2
vj
2g �M@i�
ig.
Notice that after an integration by parts
h(1 + |v|2)⌧`{q@i{�iMg}+ q�j@j[Mg]}, Mgi = �qh@i(1 + |v|2)⌧`�iMg,Mgi.139
Also, by (4.10),
�hw2(`, 2)[Ag], gi = �h(1 + |v|2)⌧`M [Ag], Mgi.
We therefore have
�hw2(`, 2)[Ag], gi =
Z
(1 + |v|2)⌧`n
�ij@j[Mg]@i[Mg] + (1� q2)�ij vi
2
vj
2[Mg]2
o
dv
�Z
w2(`, 2)@i�ig2dv
+
Z
@i(1 + |v|2)⌧`�
�ij{@j[Mg]}[Mg]� q�i[Mg]2
dv
� (1� q2)|Mg|2�,` �Z
w2(`, 2)@i�ig2dv
+
Z
@i(1 + |v|2)⌧`�
�ij{@j[Mg]}[Mg]� q�i[Mg]2
dv.
By Lemma 4.4, |@i�i| C[1 + |v|]�+1. Then as in (4.63), for m > 0, we haveZ
w2(`, 2)�
�@i�i�
� g2dv =
Z
|v|m
+
Z
|v|>m
Z
|v|m
+C
m
Z
|v|>m
(1 + |v|2)⌧`[1 + |v|]�+2[Mg]2dv
Z
|v|m
+C
m|Mg|2�,`
C(m) |�mg|2` +⌘
4|Mg|2�,` ,
(4.66)
where the last line follows from choosing m > 0 large enough. We integrate by parts
on the next term to obtainZ
@i(1 + |v|2)⌧`�ij{@j[Mg]}[Mg]dv = �1
2
Z
@j
�
@i(1 + |v|2)⌧`�ij
[Mg]2dv
By Lemma 4.4 and (4.10),
�
�@i(1 + |v|2)⌧`�i�
�+�
�@j
�
@i(1 + |v|2)⌧`�ij
�
� C(1 + |v|2)⌧`[1 + |v|]�+1.
Thus the estimate for the final term follows from the same argument as (4.66). This
establishes (4.64).
We finally establish (4.65). Notice that
|Mg|2�,` =
Z
(1 + |v|2)⌧`�
�ij@i[Mg]@j[Mg] + �ijvivj[gM ]2
dv.
140
We expand the first term in |Mg|2�,` to obtainZ
(1 + |v|2)⌧`�ij@i[Mg]@j[Mg]dv =
Z
(1 + |v|2)⌧`M2�ij@ig@jgdv
+ q2
Z
(1 + |v|2)⌧`M2�ij vi
2
vj
2g2dv
+ 2q
Z
(1 + |v|2)⌧`M2�ij vi
2{@jg}gdv
=
Z
w2(`, 2)n
�ij@ig@jg + q2�ij vi
2
vj
2g2
o
dv
+2q
Z
(1 + |v|2)⌧`M2�j{@jg}gdv.(4.67)
In the last step we used (4.10). We integrate by parts on the last term to obtain
2q
Z
(1 + |v|2)⌧`M2�j{@jg}gdv = q
Z
(1 + |v|2)⌧`M2�j{@jg2}dv
= �q
Z
(1 + |v|2)⌧`M2@j�jg2dv
� q
Z
@j(1 + |v|2)⌧`M2�jg2dv
� 2q2
Z
(1 + |v|2)⌧`M2�ij vi
2
vj
2g2dv.
Since @j(1 + |v|2)⌧` = (1 + |v|2)⌧` {2⌧`(1 + |v|2)�1vj} , we define the error as
w(v) = q�
@j�j + 2⌧`(1 + |v|2)�1�jvj
.
Then (4.67) is
=
Z
w2(`, 2)n
�ij@ig@jg � q2�ij vi
2
vj
2g2
o
dv �Z
w2(`, 2)wg2dv.
Adding the second term in |Mg|2�,` to both sides of the last display yields
|Mg|2�,` � (1� q2)|g|2�,`,2 �Z
w2(`, 2)wg2dv.
By Lemma 4.4 we have
|w(v)| C[1 + |v|]�+1.
Thus, using (4.66), for any small ⌘ > 0 we haveZ
w2(`, 2)|w|g2dv C(m)|g�m|2` +⌘
2|g|2�,`,2.
141
This completes the estimate (4.65) and the proof. ⇤
We thus conclude our estimates for the linear terms and finish the section by
estimating the nonlinear term.
Lemma 4.10. Let |↵| + |�| N , 0 # 2, q > 0 and l � 0 with ` = |�|� l. If
# = 2 restrict 0 < q < 1. Then
hw2(`,#)@↵��[g
1
, g2
], @↵� g
3
i(4.68)
CX
n
|@↵1¯�
g1
|`|@↵�↵1���1
g2
|�,`,# + |@↵1¯�
g1
|�,`|@↵�↵1���1
g2
|`,#o
|@↵� g
3
|�,`,#,
where summation is over |↵1
|+ |�1
| N, � �1
�.
Furthermore,
�
w2(`,#)@↵��[g
1
, g2
], @↵� g
3
�
(4.69)
Cn
E1/2
l (g1
)D1/2
l,# (g2
) +D1/2
l (g1
)E1/2
l,# (g2
)o
k@↵� g
3
k�,`,#.
The proof of Lemma 4.10 is more or less the same as in [38] save a few details. The
di↵erences mainly come from taking derivatives of the exponential weight w(`,#)(v)
which creates extra polynomial growth.
Proof. Recall �[g1
, g2
] in (4.46). By the product rule, we expand
hw2@↵��[g
1,g2
], @↵� g
2
i =X
C↵1↵ C�1
� ⇥G↵1�1 ,
where G↵1�1takes the form:
�hw2{�ij ⇤ @�1 [µ1/2@↵1g
1
]}@j@↵�↵1���1
g2
, @i@↵� g
3
i(4.70)
�hw2{�ij ⇤ @�1 [vi
2µ1/2@↵1g
1
]}@j@↵�↵1���1
g2
, @↵� g
3
i(4.71)
+hw2 {�ij ⇤ @�1 [µ1/2@j@
↵1g1
]}@↵�↵1���1
g2
, @i@↵� g
3
i(4.72)
+hw2{�ij ⇤ @�1 [vi
2µ1/2@j@
↵1g1
]}@↵�↵1���1
g2
, @↵� g
3
i(4.73)
�h@i[w2]{�ij ⇤ @�1 [µ
1/2@↵1g1
]}@j@↵�↵1���1
g2
, @↵� g
3
i(4.74)
+h@i[w2]{�ij ⇤ @�1 [µ
1/2@j@↵1g
1
]}@↵�↵1���1
g2
, @↵� g
3
i.(4.75)
142
The last two terms appear when we integrate by parts over the vi variable.
We estimate the last term (4.75) first. Recall from (4.51) and (4.52 ) that @i[w2] =
w2(v)w1
(v)vi where |w1
(v)| C. By first summing over i and using (4.47) we can
rewrite (4.75) as
h[w2w1
]{�ij ⇤ vi@�1 [µ1/2@j@
↵1g1
]}@↵�↵1���1
g2
, @↵� g
3
i.
Since �1 � + 2 < 0, �ij(v) 2 L2
loc(R3) and |@�1{µ1/2}| Cµ1/4, we deduce by the
Cauchy-Schwartz inequality and Lemma 4.5 that
{�ij ⇤ vi@�1 [µ1/2@j@
↵1g1
]} [|�ij|2 ⇤ µ1/8]1/2(v)X
¯��1
|µ1/32@j@↵1¯�
g1
|`.
C[1 + |v|]�+2
X
¯��1
�
�
�
@↵1¯�
g1
�
�
�
�,`,(4.76)
where we have used Lemma 2 in [38] to argue that
[|�ij|2 ⇤ µ1/8]1/2(v) C[1 + |v|]�+2.
Using the above, (4.75) is bounded by Lemma 4.5 as
CX
¯��1
�
�
�
@↵1¯�
g1
�
�
�
�,`
Z
w2(`,#)[1 + |v|]�+2|@↵�↵1���1
g2
@↵� g
3
|dv
CX
¯��1
|@↵1¯�
g1
|�,`|@↵�↵1���1
g2
|`,#|[1 + |v|]�+2@↵� g
3
|`,#
CX
¯��1
|@↵1¯�
g1
|�,`|@↵�↵1���1
g2
|`,#|@↵� g
3
|�,`,#.
This completest the estimate for (4.75).
We now estimate (4.70)-(4.74). We decompose their double integration region
[v, v0] 2 R3 ⇥ R3 into three parts:
{|v| 1}, {2|v0| � |v|, |v| � 1} and {2|v0| |v|, |v| � 1}.
Case 1. Terms (4.70)-(4.74) over {|v| 1}.143
For the first part {|v| 1}, recall �ij(v) = O(|v|�+2) 2 L2
loc. As in (4.76), we have
|�ij ⇤ @�1 [µ1/2@↵1g
1
]|+ |�ij ⇤ @�1 [viµ1/2@↵1g
1
]|+ |�ij ⇤ vi@�1 [µ1/2@↵1g
1
]|
C[1 + |v|]�+2
X
¯��
|@↵1¯�
g1
|`,
|�ij ⇤ @�1 [µ1/2@j@
↵1g1
]|+ |�ij ⇤ @�1 [{viµ1/2}@j@
↵1g1
]|
C[1 + |v|]�+2
X
¯��
|@↵1¯�
g1
|�,`.
Hence their corresponding integrands over the region {|v| 1} are bounded by
CX
�
�
�
@↵1¯�
g1
�
�
�
`[1 + |v|]�+2|@j@
↵�↵1���1
g2
|⇥
|@i@↵� g
3
|+ |@↵� g
3
|⇤
+CX
�
�
�
@↵1¯�
g1
�
�
�
�,`[1 + |v|]�+2|@↵�↵1
���1g2
|⇥
|@i@↵� g
3
|+ |@↵� g
3
|⇤
,
whose v�integral over {|v| 1} is clearly bounded by right hand side of (4.68). We
thus conclude the first part of {|v| 1} for (4.70)-(4.74).
Case 2. Terms (4.70)-(4.74) over {2|v0| � |v|, |v| � 1}.
For the second part {2|v0| � |v|, |v| � 1}, we have
|@�1µ1/2(v0)|+ |@�1{v0jµ1/2(v0)}| Cµ1/8(v0)µ1/32(v).
Thus, by the same type of estimates as in (4.76), using the region, we have
|�ij ⇤ @�1 [µ1/2@↵1g
1
]|+ |�ij ⇤ @�1 [viµ1/2@↵1g
1
]|+ |�ij ⇤ vi@�1 [µ1/2@↵1g
1
]|
C[1 + |v|]�+2µ1/64(v)X
¯��
|@↵1¯�
g1
|`,
|�ij ⇤ @�1 [µ1/2@j@
↵1g1
]|+ |�ij ⇤ @�1 [{viµ1/2}@j@
↵1g1
]|
C[1 + |v|]�+2µ1/64(v)X
¯��
|@↵1¯�
g1
|�,`.
And then the v� integrands in (4.70) to (4.74) over this region are bounded by
CX
�
�
�
@↵1¯�
g1
�
�
�
`w2(`,#)[1 + |v|]�+2µ1/64(v)|@j@
↵�↵1���1
g2
|⇥
|@i@↵� g
3
|+ |@↵� g
3
|⇤
+CX
�
�
�
@↵1¯�
g1
�
�
�
�,`w2(`,#)[1 + |v|]�+2µ1/64(v)|@↵�↵1
���1g2
|⇥
|@i@↵� g
3
|+ |@↵� g
3
|⇤
,
144
By Lemma 4.5, its v�integral is bounded by the right hand side of (4.68) because
of the fast decaying factor µ1/64(v). We thus conclude the estimate for the second
region {2|v0| � |v|, |v| � 1 } for (4.70) to (4.74).
Case 3. Terms (4.70)-(4.74) over {2|v0| |v|, |v| � 1}.
We finally consider the third part of {2|v0| |v|, |v| � 1}, for which we shall
estimate each term from (4.70) to (4.74). The key is to taylor expand �ij(v � v0). To
estimate (4.70) over the this region we expand �ij(v � v0) to get
(4.77) �ij(v � v0) = �ij(v)�X
k
@k�ij(v)v0k +
1
2
X
k,l
@kl�ij(v)v0kv
0l.
where v is between v and v � v0. We plug (4.77) into the integrand of (4.70). From
(4.40), (4.41) and (4.47), we can decompose @j@↵�↵1���1
g2
and @i@↵� g
3
into their Pv parts
as well as I � Pv parts. For the first term in the expansion (4.77) we have
X
ij
�ij(v)@j@↵�↵1���1
g2
(v)@i@↵� g
3
(v)
=X
ij
�ij(v){[I � Pv]@j@↵�↵1���1
g2
(v)}{[I � Pv]@i@↵� g
3
(v)}.
Here we have used (4.47) so that sum of terms with either Pv@j@↵�↵1���1
g2
or Pv@i@↵� g
3
vanishes. The absolute value of this is bounded by
(4.78) C[1 + |v|]�+2|[I � Pv]@j@↵�↵1���1
g2
(v)|⇥ |[I � Pv]@i@↵� g
3
(v)|.
For the second term in the expansion (4.77), by taking a k derivative of
X
i,j
�ij(v)vivj = 0
we have
X
i,j
@k�ij(v)vivj = �2
X
j
�kj(v)vj = 0.
Therefore
X
i,j
@k�ij(v)Pv@j@
↵�↵1���1
g2
Pv@i@↵� g
3
= 0.
145
Splitting @j@↵�↵1���1
g2
and @i@↵� g
3
into their Pv and I � Pv parts yields
X
i,j
@k�ij(v)@j@
↵�↵1���1
g2
(v)@i@↵� g
3
(v)
=X
i,j
@k�ij(v){[Pv@j@
↵�↵1���1
g2
][I � Pv]@i@↵� g
3
+ [I � Pv]@j@↵�↵1���1
g2
[Pv@i@↵� g
3
]}
+X
i,j
@k�ij(v)[I � Pv]@j@
↵�↵1���1
g2
[I � Pv]@i@↵� g
3
.
Notice that |@k�ij(v)| C[1 + |v|]�+1 for |v| � 1, we therefore majorize the above by
C[1 + |v|]�/2{|Pv@j@↵�↵1���1
g2
|+ |Pv@i@↵� g
3
|}
⇥[1 + |v|](�+2)/2{|[I � Pv]@j@↵�↵1���1
g2
|+ |[I � Pv]@i@↵� g
3
|}(4.79)
+C[1 + |v|]�+1|[I � Pv]@j@↵�↵1���1
g2
||[I � Pv]@i@↵� g
3
|.
Next, we estimate third term in (4.77). Using the region we have
(4.80)1
2|v| |v|� |v0| |v| |v0|+ |v| 3
2|v|.
Thus
|@kl�ij(v)| C[1 + |v|]�,
and
(4.81)�
�@kl�ij(v)@j@
↵�↵1���1
g2
(v)@i@↵� g
3
(v)�
� C[1 + |v|]�|@j@↵�↵1���1
g2
@i@↵� g
3
|.
Combining (4.77), (4.78), (4.79) and (4.81) we obtain the estimate�
�
�
�
�
X
i,j
�ij(v � v0)@i@↵�↵1���1
g2
@i@↵� g
3
�
�
�
�
�
C[1 + |v0|]2�
�
�
�
�
X
i,j
�ij(v)@j@↵�↵1���1
g2
(v)@i@↵� g
3
(v)
�
�
�
�
�
+C[1 + |v0|]2�
�
�
�
�
X
i,j
@k�ij(v)@j@
↵�↵1���1
g2
(v)@i@↵� g
3
(v)
�
�
�
�
�
+C[1 + |v0|]2X
i,j
�
�@kl�ij(v)@j@
↵�↵1���1
g2
(v)@i@↵� g
3
(v)�
�
C[1 + |v0|]2{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ij@i@↵� g
3
@j@↵� g
3
}1/2,
146
where we have used (4.42) in the last line. The v integrand over {2|v0| |v|, |v| � 1}
in (4.70) is thus bounded by
w2
Z
[1 + |v0|]2µ1/4(v0)|@↵1¯�
g1
(v0)|dv0
⇥{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ij@i@↵� g
3
@j@↵� g
3
}1/2
C|@↵¯� g
1
|`{w2�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{w2�ij@i@↵� g
3
@j@↵� g
3
}1/2.
Its further integration over v is bounded by the right hand side of (4.68).
We now consider the second term (4.71). We again expand �ij(v � v0) as
(4.82) �ij(v � v0) = �ij(v)�X
k
@k�ij(v)v0k,
with v between v and v � v0. SinceP
j �ij(v)vj = 0 we obtain as before
X
j
�ij(v)@j@↵�↵1���1
g2
(v)@↵� g
3
(v) =X
j
�ij(v){I � Pv}@j@↵�↵1���1
g2
(v)⇥ @↵� g
3
(v)
C[1 + |v|]�+2|{I � Pv}@j@↵�↵1���1
g2
(v)||@↵� g
3
(v)|(4.83)
C|[1 + |v|] �+22 {I � Pv}@j@
↵�↵1���1
g2
(v)||[1 + |v|] �+22 @↵
� g3
(v)|
C{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ijvivj|@↵� g
3
|2}1/2,
where we have used (4.42). From (4.80), |@k�ij(v)| C[1 + |v|]�+1. Hence
|@k�ij(v)@j@
↵�↵1���1
g2
(v)@↵� g
3
(v)|(4.84)
C[1 + |v|]�+1{|@j@↵�↵1���1
g2
(v)|}{|@↵� g
3
(v)|}
C{[1 + |v|]�/2|@j@↵�↵1���1
g2
(v)|}{[1 + |v|] �+22 |@↵
� g3
(v)|}
C{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ijvivj|@↵� g
3
|2}1/2.
From (4.83) and (4.84), we thus conclude�
�
�
�
�
X
ij
�ij(v � v0)@j@↵�↵1���1
g2
(v)@↵� g
3
(v)
�
�
�
�
�
C[1 + |v0|]{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ijvivj|@↵� g
3
|2}1/2.
147
We thus conclude that the v integrand in (4.71) can be majorized by
CX
Z
[1 + |v0|]µ1/4(v0)|@↵1¯�
g1
(v0)|dv0
⇥w2{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ij@i@↵� g
3
@j@↵� g
3
}1/2
CX
|@↵1¯�
g1
|`{w2�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{w2�ij@i@↵� g
3
@j@↵� g
3
}1/2.
Further integration over v shows that this bounded by the right hand side of (4.68).
We now consider the third term (4.72) over {2|v0| |v|, |v| � 1}. We use an
integration by parts inside the convolution to split (4.72) into two parts
(4.85) �ij ⇤ @�1 [µ1/2@j@
↵1g1
] = @j�ij ⇤ @�1 [µ
1/2@↵1g1
]� �ij ⇤ @�1 [@jµ1/2 @↵1g
1
].
Recall expansion (4.82) and decompose
@i@↵� g
3
= Pv@i@↵� g
3
+ [I � Pv]@i@↵� g
3
.
By similar estimates to (4.83) and (4.84), the second part of (4.72) can be estimated
asZ
{|v|�1,2|v0||v|}|w2 �ij(v � v0)@�1 [@jµ
1/2(v0)@↵1g1
(v0)]@↵�↵1���1
g2
(v)@i@↵� g
3
(v)|
=
Z
|v|�1,2|v0||v||w2[�ij(v)� @k�
ij(v)v0k]@�1 [@jµ1/2(v0)@↵1g
1
(v0)]@↵�↵1���1
g2
@i@↵� g
3
|
C�
�
�
@↵1¯�
g1
�
�
�
`
�
�
�
[1 + |v|] �+22 @↵�↵1
���1g2
�
�
�
`,#
�
�
�
w#[1 + |v|] �+22 {I � Pv}@i@
↵� g
3
�
�
�
`,#
+C�
�
�
@↵1¯�
g1
�
�
�
`
�
�
�
[1 + |v|] �+22 @↵�↵1
���1g2
�
�
�
`,#
�
�[1 + |v|]�/2@i@↵� g
3
�
�
`,#.
By Lemma 4.5, this is bounded by the right hand side of (4.68).
For the first part of (4.72) by (4.85) notice that our integration region implies
|@j�ij(v � v0)| C[1 + |v|]�+1.
We thus haveZ
{|v|�1,2|v0||v|}w2 |@j�
ij(v � v0)|@�1 [µ1/2(v0)@↵1g
1
(v0)]@↵�↵1���1
g2
(v)@i@↵� g
3
(v)
CX
¯��1
�
�
�
@↵1¯�
g1
�
�
�
`
�
�
�
[1 + |v|] �+22 @↵�↵1
���1g2
�
�
�
`,#
�
�[1 + |v|]�/2@j@↵� g
3
�
�
`,#,
148
which is bounded by the right hand side of (4.68) by Lemma 4.5.
Next consider (4.73) over {2|v0| |v|, |v| � 1}. We split (4.73) as in (4.85)
�ij ⇤ @�1 [viµ1/2@j@
↵1g1
] = @j�ij ⇤ @�1 [viµ
1/2@↵1g1
]� �ij ⇤ @�1 [@j{viµ1/2} @↵1g
1
].
Since |�ij(v�v0)| C[1+ |v|]�+2, and |@j�ij(v�v0)| C[1+ |v|]�+1, (4.73) is bounded
by
Z
w2[1 + |v|]�+1|@�1 [v0iµ
1/2(v0)@↵1g1
(v0)]@↵�↵1���1
g2
@↵� g
3
|dv0dv
+
Z
w2[1 + |v|]�+2|@�1 [@j{v0iµ1/2(v0)}@↵1g1
(v0)]@↵�↵1���1
g2
@↵� g
3
|dv0dv
CX
¯��1
�
�
�
@↵1¯�
g1
�
�
�
`
�
�@↵�↵1���1
g2
�
�
�,`,#
�
�@↵� g
3
�
�
�,`,#.
We thus conclude the estimate for (4.73).
Finally, consider the term (4.74) over {2|v0| |v|, |v| � 1}. First sum over vi so
that (4.74) is given by
h[w2w1
]{�ij ⇤ vi@�1 [µ1/2@↵1g
1
]}@j@↵�↵1���1
g2
, @↵� g
3
i
We again expand �ij(v� v0) as in (4.82). By (4.47), we have the estimates (4.83) and
(4.84). Plugging (4.83) and (4.84) into (4.82), we thus conclude that the v integrand
in (4.74) can be majorized by
CX
Z
[1 + |v0|]2µ1/4(v0)|@↵1¯�
g1
(v0)|dv0
⇥w2{�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{�ijvivj|@↵� g
3
|2}1/2
CX
|@↵1¯�
g1
|`{w2�ij@i@↵�↵1���1
g2
@j@↵�↵1���1
g2
}1/2{w2�ijvivj|@↵� g
3
|2}1/2.
By further integrating over v, this bounded by the right hand side of (4.68). And
thus, the proof of (4.68) is complete.
149
The proof of (4.69) now follows from the Sobolev embedding H2(T3) ⇢ L1(T3)
and (4.68). Without loss of generality, assume |↵1
|+ |�| N/2 in (4.68). Then✓
supx|@↵1
¯�g1
(x)|`◆
|@↵�↵1���1
g2
|�,`,# +
✓
supx|@↵1
¯�g1
(x)|�,`
◆
|@↵�↵1���1
g2
|`,#
⇣
X
k@↵0
�0 g1
k`
⌘
|@↵�↵1���1
g2
|�,`,# +⇣
X
k@↵0
�0 g1
k�,`
⌘
|@↵�↵1���1
g2
|`,#,
where the summation is over |↵0|+ |�0| N2
+2 N . We deduce (4.69) by integrating
(4.68) over T3 and using the this computation. ⇤
4.5. Energy Estimate and Global Existence
In this section we will prove the energy estimate which is a crucial step in con-
structing global solutions. By now, it is standard to prove local existence of small
solutions using the estimates either in Section 4.3 for the Boltzmann case or Section
4.4 for the Landau case:
Theorem 4.3. For any su�ciently small M⇤ > 0, T ⇤ > 0 with T ⇤ M⇤
2
and
1
2
X
|↵|+|�|N
||@↵� f
0
||2|�|�l,# M⇤
2,
there is a unique classical solution f(t, x, v) to (4.5) in either the Boltzmann or the
Landau case in [0, T ⇤)⇥ T3 ⇥ R3 such that
sup0tT ⇤
8
<
:
1
2
X
|↵|+|�|N
||@↵� f ||2|�|�l,#(t) +
Z t
0
Dl,#(f)(s)ds
9
=
;
M⇤,
and 1
2
P
|↵|+|�|N ||@↵� f ||2|�|�l,#(t) +
R t
0
Dl,#(f)(s)ds is continuous over [0, T ⇤).
Next, we define some notation. For fixed N � 8, 0 m N and #, q, l � 0, a
modified instant energy functional satisfies
1
CEm
l,#(g)(t) X
|�|m,|↵|+|�|N
||@↵� g(t)||2|�|�l,# CEm
l,#(g)(t).(4.86)
Similarly the modified dissipation rate is given by
Dml,#(g)(t) ⌘
X
|�|m,|↵|+|�|N
||@↵� g(t)||2D,|�|�l,#.(4.87)
150
Note that, ENl,#(g)(t) = El,#(g)(t) and DN
l,#(g)(t) = Dl,#(g)(t). And as before, we will
write Eml,0(g)(t) = Em
l (g)(t) and Dml,0(g)(t) = Dm
l (g)(t). Now we are ready to state a
result from equation (4.5) in [60] using this new notation:
Lemma 4.11. Let f(t, x, v) be a classical solution to (4.5) satisfying (4.13) in
either the Boltzmann or the Landau case. In the Boltzmann case assume ⌧ �
but in the Landau case assume ⌧ �1 in (4.10). For any l � 0, there exists Ml,
�l = �l(Ml) > 0 such that if
(4.88)1
2
X
|↵|+|�|N
||@↵� f ||2|�|�l(t) Ml,
then for any 0 m N we have an instant energy functional such that
d
dtEm
l (f)(t) +Dml (f)(t) C
p
El(f)(t)Dl(f)(t).(4.89)
We will bootstrap this energy estimate without an exponential weight (# = 0) to
Corollary 4.1 in the Boltzmann case and Lemma 4.9 on the Landau case to obtain
the following general energy estimate.
Lemma 4.12. Fix N � 8, 0 < # 2, q > 0 and l � 0. If # = 2 let 0 < q < 1. In
the Boltzmann case assume ⌧ � but in the Landau case assume ⌧ �1 in (4.10).
Let f(t, x, v) be a classical solution to (4.5) satisfying (4.13) and (4.88) in either the
Boltzmann or the Landau case. For any given 0 m N there is a modified instant
energy functional such that
d
dtEm
l,#(f)(t) +Dml,#(f)(t) CE1/2
l,# (f)(t)Dl,#(f)(t).(4.90)
Proof. We use an induction over m, the order of the v�derivatives. For m = 0,
by taking the pure @↵ derivatives of (4.5) we obtain
{@t + v ·rx} @↵f + L{@↵f} =X
↵1↵
C↵1↵ �(@↵1f,@
↵�↵1f)(4.91)
151
Multiply w2(�l,#)@↵f with (4.91), integrate over T3 ⇥ R3 and sum over |↵| N to
deduce the following for some constant C > 0,
X
|↵|N
⇢
1
2
d
dtk@↵f(t)k2
�l,# +�
w2(�l,#)L{@↵f(t)}, @↵f(t)�
�
CE1/2
l,# (f)(t)Dl,#(f)(t).(4.92)
We have used Lemma 4.3 in the Boltzmann case and (4.69) in the Landau case to
bound the r.h.s. of (4.91). Notice that Lemma 4.2 (in the Boltzmann case) implies
�
w2(�l,#)L{@↵f(t)}, @↵f(t)�
= k@↵f(t)k2
⌫,�l,# ��
w2(�l,#)K{@↵f(t)}, @↵f(t)�
� 1
2k@↵f(t)k2
⌫,�l,# � Ck@↵f(t)k2
⌫,�l,
where C > 0 is a large constant. In the Landau case, Lemma 4.9 gives
�
w2(�l,#)L{@↵f(t)}, @↵f(t)�
� �qk@↵f(t)k2
�,�l,# � Ck�C@↵f(t)k2
�l
� �qk@↵f(t)k2
�,�l,# � Ck@↵f(t)k2
�,�l.
Without loss of generality assume 0 < �q < 1
2
. Then in either case, plugging these
into (4.92) we have
X
|↵|N
⇢
1
2
d
dtk@↵f(t)k2
�l,# +1
2k@↵f(t)k2
D,�l,# � Ck@↵f(t)k2
D,�l
�
Cq
El,#(f)(t)Dl,#(f)(t).
Add to this inequality to (4.89) with m = 0, possibly multiplied by a large constant,
to obtain (4.90) with m = 0.
Now assume the Lemma is valid for some fixed m > 0. For |�| = m + 1, taking
@↵� of (4.5) we obtain
{@t + v ·rx} @↵� f + @�{L@↵f}(4.93)
= �X
|�1|=1
C�1� @�1v ·rx@
↵���1
f +X
C↵1↵ @��(@↵1f,@
↵�↵1f).
We take the inner product of (4.93) with w2(|�| � l,#)@↵� f over T3 ⇥ R3. The first
inner product on the left is equal to 1
2
ddt||@↵
� f(t)||2|�|�l,#. From either Corollary 4.1
152
in the Boltzmann case or Lemma 4.9 in the Landau case, we deduce that the inner
product of @�{L@↵f} is bounded from below as
X
|�|=m+1
�
w2@�{L@↵f}, @↵� f�
� 1
2
X
|�|=m+1
||@↵� f ||2D,|�|�l,# � C
X
|¯�|m
||@↵¯� f ||2D,|¯�|�l,#.
Since |�1
| = 1, as in [37] the streaming term on the r.h.s. of (4.93) is bounded by
(w2(|�|� l,#){@�1vj}@xj
@↵���1
f, @↵� f)
Z
w2(|�|� l,#)|@xj
@↵���1
f@↵� f |dxdv
�
�w(|�|+ 1/2� l,#)@↵� f�
�
�
�w(1/2 + {|�|� 1}� l,#)@xj
@↵���1
f�
�
⌘�
�@↵� f�
�
2
D,|�|�l,#+ C⌘
�
�@xj
@↵���1
f�
�
2
D,|���1|�l,#.
Further, by Lemma 4.3 in the Boltzmann case and Lemma 4.10 in the Landau case,
the inner product involving � on the r.h.s. of (4.93) is CE1/2
l,# (f)(t)Dl,#(f)(t).
Collect terms and sum over |�| = m + 1, |↵|+ |�| N to obtain
X
|�|=m+1,|↵|+|�|N
⇢
1
2
d
dt||@↵
� f(t)||2|�|�l,# +
✓
1
2�W⌘
◆
�
�@↵� f�
�
2
D,|�|�l,#
�
X
|�|=m+1,|↵|+|�|N
C
0
@
X
|�1|=1
�
�@xj
@↵���1
f�
�
2
D,|���1|�l,#+X
|¯�|m
||@↵¯� f ||2D,|�|�l,#
1
A
+CZm+1
E1/2
l,# (f)(t)Dl,#(f)(t).
Here Zm+1
=P
|�|=m+1,|↵|+|�|N 1 and W =P
|�1|=1
C�1� . Choosing ⌘ > 0 such that
1
2
�W⌘ = 1
4
> 0 we get
X
|�|=m+1,|↵|+|�|N
⇢
1
2
d
dt||@↵
� f(t)||2|�|�l,# +1
4
�
�@↵� f�
�
2
D,|�|�l,#
�
CX
|�|m,|↵|+|�|N
||@↵� f ||2D,|�|�l,# + CZm+1
q
E l,#(f)(t)Dl,#(f)(t).
Choose Am+1
such that Am+1
� C � 1. Now multiply (4.90) for |�| m by Am+1
and add it to the display above to obtain (4.90) for |�| m + 1. We thus conclude
the energy estimate. ⇤
With Lemma 4.12, we can prove existence of global in time solutions with an
exponential weight using exactly the same argument as in the last Section of [39].
153
4.6. Proof of Exponential Decay
In this section we prove exponential decay using the di↵erential inequality (4.14)
and the uniform bound (4.15) with # > 0. The main di�culty in establishing decay
from (4.14) is rooted in the fact that the dissipation Dl,#(f)(t) is in general weaker
than the instant energy El,#(f)(t). As in the work of Caflisch [8], the key point is to
split El(f)(t) into a time dependent low velocity part
E = {|v| ⇢tp0},
and it’s complementary high velocity part Ec = {|v| > ⇢tp0}, where p0 > 0 and ⇢ > 0
will be chosen at the end of the proof.
First consider the Boltzmann case. Let E lowl (f)(t) be the instant energy restricted
to E. Then from (4.16), for t > 0, we have
(4.94) Dl(f)(t) � C⇢�t�p0E lowl (f)(t).
Plugging this into the the di↵erential inequality (4.14) we obtain
d
dtEl(f)(t) + C⇢�t�p0E low
l (f)(t) 0.
Letting Ehighl (f)(t) = El(f)(t)� E low
l (f)(t) we have
d
dtEl(f)(t) + C⇢�t�p0El(f)(t) C⇢�t�p0Ehigh
l (f)(t).
Define � = C⇢�/p where for now p = �p0 + 1 and p0 > 0 is arbitrary. Then
d
dtEl(f)(t) + �ptp�1El(f)(t) �ptp�1Ehigh
l (f)(t).
Equivalently
d
dt
�
e�tpEl(f)(t)�
�ptp�1e�tpEhighl (f)(t).
The integrated form is
El(f)(t) e��tpEl(f0
) + �pe��tpZ t
0
sp�1e�spEhighl (f)(s)ds.
154
Above p > 0 or equivalently �p0 > �1 is assumed to guarantee the integral on the
r.h.s. is finite. Since Ehighl (f)(s) is on Ec = {|v| > ⇢sp0}
Ehighl (f)(s) = Ehigh
l,0 (f)(s) Ce�q
2⇢s#p
0Ehigh
l,# (f)(s).
In the last display we have used the region and
1 exp⇣q
2(1 + |v|2)#
2
⌘
e�q
2 |v|# exp⇣q
2(1 + |v|2)#
2
⌘
e�q
2⇢s#p
0.
Hence (4.15) implies
El(f)(t) e��tp✓
El,0(f0
) + �pEl,#(f0
)
Z t
0
sp�1e�sp� q
2⇢s#p
0ds
◆
.
The biggest exponent p that we can allow with this splitting is p = #p0; since also
p = �p0 + 1 we have p0 = 1
#��so that
p =�
#� �+ 1 =
#
#� �.
Further choose ⇢ > 0 large enough so that � = C⇢�/p < q2
⇢ (� < 0) and henceZ 1
0
sp�1e�sp� q
2⇢sp
ds < +1.
This completes the proof of decay in the Boltzmann case.
For the proof of decay in the Landau case, instead of (4.94), we use Lemma 4.5
to see that
Dl(f)(t) � C⇢2+�t(2+�)p0E lowl (f)(t).
And the rest of the proof is exactly the same. But we find, in this case, that p =
##�(2+�)
. Q.E.D.
155
CHAPTER 5
On the Relativistic Boltzmann Equation
In this chapter, we will discuss a few basic mathematical properties of the relativis-
tic Boltzmann equation. The relativistic Boltzmann equation is the central equation
in relativistic collisional kinetic theory. In the next few paragraphs, we will review
most of the mathematical theory of the relativistic Boltzmann equation. Books which
discuss relativistic Kinetic theory include [13,15,30,59,63].
Lichnerowicz and Marrot [49] are said to be the first to write down the full rela-
tivistic Boltzmann equation, including collisional e↵ects, in 1940. In 1967, Bichteler
[6] showed that the general relativistic Boltzmann equation has a local solution if the
initial distribution function decays exponentially with the energy and if the di↵er-
ential cross-section is bounded. Dudynski and Ekiel-Jezewska [25], in 1988, showed
that the linearized equation admits unique solutions in L2. Afterwards, Dudynski [24]
studied the long time and small-mean-free-path limits of these solutions. Then, in
1992, Dudynski and Ekiel-Jezewska [26] proved global existence of large data Diperna-
Lions solutions [23]. In this paper, they assume the relativistic Boltzmann equation
is causal (i.e. solutions depend on the inital data only inside the past interior of the
light cone), a result which they had previously established [27, 28]. This result has
since been extended in [44,45].
In 1991, Glassey and Strauss [31] studied the collision map that carries the pre-
collisional momentum of a pair of colliding particles into their momentum post-
collision. The collisional map becomes very important as soon as one wants to study
uniqueness and regularity of the full non-linear equation. In 1993, Glassey and Strauss
[32] proved existence and uniqueness of smooth solutions which are initially close to
a relativistic Maxwellian and in a periodic box. They also established exponential
156
convergence to Maxwellian. In 1995, they extended these results to the whole space
case [33], where the convergence rate is polynomial.
In 1996, Andreasson [1] showed that the gain term is regularizing. This is a
generalization of Lions [51, 52] result in the non-relativistic case. In 1997, Wennberg
[71] proved the regularity of the gain term for both the relativistic and non-relativistic
case in a unified framework. In 2004, Calogero [10] proved existence of local-in-time
solutions independent of the speed of light and established a rigorous Newtonian limit.
In the same year Andreasson, Calogero and Illner [2] showed that removal of the loss
term for the Boltzmann equation (relativistic or not) leads to finite time blow up of
a solution.
Our goal in this chapter is to define carefully many aspects of the relativistic
Boltzmann equation. We will discuss two di↵erent representations for the collsion
operator. And then we write down four examples of di↵erential cross sections in
the relativistic case. After that we develop three Lorentz transformations which are
useful in relativistic collisional kinetic theory. We then rigorously derive the di↵erent
representations for the collision operator. We then derive the Hilbert-Schmidt form
for the linearized collision operator. And we finish this chapter with some remarks
on the relativistic Vlasov-Maxwell-Boltzmann system.
Much of this chapter is expository. However most of this material is scattered
and/or written in the language of physics. Our motivation is to present all of this
material in one place and in one language. The only thing that is possibly original
in this chapter is the precise form of two of the Lorentz Transformations. These may
be useful in future mathematical investigations.
5.1. The Relativistic Boltzmann Equation
The relativisitic Boltzmann Equation is
(5.1) @tF + p ·rxF = C(F, F ), F (0, x, p) = F0
(x, p).
Here F = F (t, x, p) is a spatially periodic function of time t 2 [0,1), space x 2 ⌦ ⇢
R3 and momentum p 2 R3. Let c denote the speed of light. the energy of a relativistic
157
particle with momentum p is defined by p0
=p
c2 + |p|2 and the normalized velocity
of a particle by
p =p
p
1 + |p|2/c2
.
Moreover, the relativistic Boltzmann collision operator is commonly written in the
physics literature [7, 15] as
(5.2) C(F, G) =c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
W (p, q|p0, q0)[F (p0)G(q0)� F (p)G(q)].
Here the transition rate, W (p, q|p0, q0), has the form
(5.3) W (p, q|p0, q0) = s�(g, ✓)�(4)(P + Q� P 0 �Q0),
where � is called the di↵erential cross-section. Denote the relativistic four vector, P ,
by P = (p0
, p)t. Q = (q0
, q)t is another four-vector. The post-collisional momentum,
(p0, q0), and energy, (p00
, q00
), satisfy:
(5.4) P + Q = P 0 + Q0.
Denote the Lorentz inner product for 4-vectors as
P ·Q = p0
q0
�3
X
i=1
piqi.
With this notation in hand, we write
s = s(P, Q) ⌘ |P + Q|2 ⌘ (P + Q) · (P + Q) = 2�
P ·Q + c2
�
,(5.5)
where s is the normalized total energy in the center-of-momentum system:
(5.6) p + q = 0.
Also, 2g is called the relative momentum, where
g = g(P, Q) ⌘ |P �Q| ⌘p
�(P �Q) · (P �Q) =p
2 (P ·Q� c2).(5.7)
Note that
(5.8) s = 2�
P ·Q + c2
�
= 2�
P ·Q� c2
�
+ 4c2 = g2 + 4c2.
158
Now (5.4) and (5.5) together imply that s and therefore g are collision invariants:
s(P, Q) = s(P 0, Q0) and g(P, Q) = g(P 0, Q0).
Next, the angle ✓–which is the scattering angle in the center-of-momentum system
(5.6)–is defined by
cos ✓ ⌘ �(P �Q) · (P 0 �Q0)/g2.(5.9)
We remark that in general the expression in (5.9) is not well-defined because the
r.h.s. could be greater than one. But the conservation of momentum and energy
(5.4) makes this a good definition [30, p.113].
A smooth solution to the relativistic Boltzmann equation, (5.1), which decays
su�ciently rapidly satisfies the conservation of mass, momentum and energy:
d
dt
Z
T3⇥R3
F (t)dxdp =d
dt
Z
T3⇥R3
pF (t)dxdp =d
dt
Z
T3⇥R3
p0
F (t)dxdp = 0.
Boltzmann’s famous H-Theorem for the relativistic Boltzmann equation (5.1) is
d
dt
Z
T3⇥R3
F (t) ln F (t)dxdp 0.
We write the relativistic Maxwellian or Juttner solution as
(5.10) J(p) ⌘ exp (�cp0
)
4⇡cK2
(c2).
Since C(J, J) ⌘ 0 this is a steady state solution to (5.1). Here K2
is a bessel function
defined by the restrictionR
R3 J(p)dp = 1 [48, p.449].
In the literature, there are two ways to carry out the delta function integrations
in the collision operator. These are analagous to two well known expressions for the
post-collisional velocities in the classical (non-relativistic) Boltzmann theory.
5.1.1. Center-of-Momentum Collision Operator. Following [7, 15] four of
the integrations can be carried out in the center-of-momentum system to get rid of
the delta functions and obtain
(5.11) C(F, G) =
Z
R3⇥S2
v�(g, ✓)[F (p0)G(q0)� F (p)G(q)]d!dq.
159
where v = v(p, q) is the Møller velocity given by
(5.12) v = v(p, q) ⌘ c
2
s
�
�
�
�
p
p0
� q
q0
�
�
�
�
2
� 1
c2
�
�
�
�
p
p0
⇥ q
q0
�
�
�
�
2
=c
4
gp
s
p0
q0
.
The post collisional momentum in the expression (5.11) can be written:
p0i =pi + qi
2+ g
✓
!i + (� � 1)(pi + qi)(p + q) · !|p + q|2
◆
,
q0i =pi + qi
2� g
✓
!i + (� � 1)(pi + qi)(p + q) · !|p + q|2
◆
,
(5.13)
where � = (p0
+ q0
)/p
s. The energies are then
p00
=p
0
+ q0
2+
gps! · (p + q),
q00
=p
0
+ q0
2� gp
s! · (p + q).
These clearly satisfy (5.4). Further
cos ✓ = k · !
where k is a unit vector. Then one can write d! = sin ✓d✓d with 0 ✓ ⇡ and
0 2⇡. We will perform this computation in Section 5.4. In fact, we will show
in Section 5.4 that (P 0, Q0) can be defined more generally only up to a (non-unique)
Lorentz Transformation.
This expression is the relativistic analogoue of the center-of-mass variables for the
classical Boltzmann collision operator. In the non-relativistic case, the center-of-mass
variables are
p0 =p + q
2+
1
2|p� q|!,
q0 ! p + q
2� 1
2|p� q|!,
And the collision operator with hard-sphere interaction is
C1(F, G) =1
2
Z
R3⇥S2
|p� q|[F (p0)G(q0)� F (p)G(q)]d!dq.
Indeed, when � = 1, the formal Newtonian limit (c " 1) of (5.11) with variables
(5.13) is this center-of-mass system.
160
5.1.2. Another expression for the Collision Operator. Glassey and Strauss
showed in [32] that another reduction can be performed, without using the center-of-
momentum system, to obtain
(5.14) C(F, G) =
Z
R3⇥S2
s�(g, ✓)
p0
q0
B(p, q,!)[F (p0)G(q0)� F (p)G(q)]d!dq,
where the kernel is
(5.15) B(p, q,!) ⌘ c(p
0
+ q0
)2p0
q0
�
�
�
! ·⇣
pp0� q
q0
⌘
�
�
�
[(p0
+ q0
)2 � (! · [p + q])2]2.
In this expression, the post collisional momentum’s are given by
p0 = p + a(p, q,!)!,
q0 = q � a(p, q,!)!,(5.16)
where
a(p, q,!) =2(p
0
+ q0
)p0
q0
n
! ·⇣
qq0� p
p0
⌘o
(p0
+ q0
)2 � {! · (p + q)}2
.
And the energies can be expressed as p00
= p0
+ N0
and q00
= q0
�N0
with
N0
⌘ 2! · (p + q){p0
(! · q)� q0
(! · p)}(p
0
+ q0
)2 � {! · (p + q)}2
.
These expressions clearly satisfy the collisional conservations (5.4). The angle is then
defined by plugging these into (5.9). The Jacobian for the transformation (p, q) !
(p0, q0) in these variables [31] is
(5.17)@(p0, q0)@(p, q)
= �p00
q00
p0
q0
.
Moreover we have the following quanities which are invariant before and after colli-
sions:
p0
q0
! ·✓
q
q0
� p
p0
◆�
= p00
q00
! ·✓
p0
p00
� q0
q00
◆�
,
P ·Q = P 0 ·Q0.
Therefore for fixed ! 2 S2 we have
(5.18) B(p, q,!) = B(p0, q0,!).
161
One can use these to derive further identities.
Now these variables are analogous to the other common non-relativistic Boltzmann
variables
p0 = p + ! · (q � p)!,
q0 = q � ! · (q � p)!.
With these post-collisional velocities, the non-relativistic collision operator with hard-
sphere interaction is
C1(F, G) =
Z
R3⇥S2
|! · (p� q)| [F (p0)G(q0)� F (p)G(q)]d!dq.
And when � = 1 the formal Newtonian limit of (5.14) with variables (5.16) is this
standard hard-sphere collision operator.
5.2. Examples of relativistic Boltzmann cross-sections
The calculation of the di↵erential cross-section is in the relativistic situation utilzes
quantum field theory [55]. In the mathematical literature of the relativistic Boltzmann
equation it is hard to find precise physically relevant examples of these di↵erential
cross-sections. In this section we write down a few examples of di↵erential cross
sections. However, I am not a physicist and I am not claiming these are the physically
relevant examples.
Above, we have written everything down with the mass m normalized to one,
m = 1. Here we write down the mass before normalization.
5.2.0.1. Short Range Interactions. [29, 55] For short range interactions,
s� ⌘ constant.
This is the relativistic analouge of the hard-sphere cross-section in the Newtonian
case. Indeed, as we have already mentioned, the Newtonian limit of the relativistic
Boltzmann equation in this case is the hard-sphere Boltzmann equation.
162
5.2.0.2. Møller Scattering. [15, p.350] Møller scattering is used as an approxima-
tion of electron-electron scatttering.
�(g,⇥) = r2
0
1
u2(u2 � 1)2
⇢
(2u2 � 1)2
sin4⇥�
2u4 � u2 � 1
4
sin2⇥+
1
4(u2 � 1)2
�
.
where the magnitude of total four-momentum scaled w.r.t. total mass is
u =
ps
2mc
and r0
= e2
4⇡mc2is the classical electron radius. Photon-photon scattering is often
neglected because the size of the cross-section is ‘negligible’.
5.2.0.3. Compton Scattering. [15, p.351] Compton scattering is an approximation
of photon-electron scattering.
�(g,⇥) =1
2r2
0
(1� ⇠)
(
1 +1
4
⇠2(1� cos⇥)2
1� 1
2
⇠(1� cos⇥)+
✓
1� (1� 1
2
⇠)(1� cos⇥)
1� 1
2
⇠(1� cos⇥)
◆
2
)
,
where
⇠ = 1� m2c2
s.
See [29, p.81] for the Newtonian limit in this case.
5.2.0.4. (elastic) Neutrino Gas. [22, p.478] For a neutrino gas, the di↵erential
cross section is independent of the scattering a angle ⇥.
�(g,⇥) =G2
⇡~2c2
g2,
where G is the weak coupling constant and 2⇡~ is Plank’s constant. These are massless
particles! See also [15, p.290].
5.2.0.5. Israel particles. [43, p.1173] The Isreal particles are the analogue of the
“maxwell molecules” cross section in the Newtonian theory.
�(g,⇥) =m
2g
b(⇥)
1 + (g/mc)2
.
With this cross section, Israel derives eigenfunctions for the Linearized relativistic
Boltzmann collision operator. Variants of this cross section are used in [13, 56]. For
instance the cross section for “Maxwell Particles” is formed by removing the factor
163
1 + (g/mc)2. Note that it converges to the maxwell molecules cross section in the
Newtonian limit.
5.3. Lorentz Transformations
In this section, we will write down three Lorentz Transformations which can be
useful in relativistic Kinetic Theory. Let ⇤ be a 4⇥ 4 matrix denoted by
⇤ = (⇤ij)0i,j3
=
0
B
B
B
B
B
B
@
⇤0
⇤1
⇤2
⇤3
1
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
@
⇤0
0
⇤0
1
⇤0
2
⇤0
3
⇤1
0
⇤1
1
⇤1
2
⇤1
3
⇤2
0
⇤2
1
⇤2
2
⇤2
3
⇤3
0
⇤3
1
⇤3
2
⇤3
3
1
C
C
C
C
C
C
A
= (⇤0
⇤1
⇤2
⇤3
) .
Here ⇤i (i = 0, 1, 2, 3) are used to denote the row’s and ⇤i are used to denote the
columns’s of the matrix. We assume the entries of the matrix are real. For the basics
of Lorentz Transformations, we refer to [14,54,70].
Definition 5.1. ⇤ is a Lorentz Transformation if
(⇤P ) · (⇤Q) = P ·Q
Let D = diag(1,�1,�1,�1). Equivalently, ⇤ is a Lorentz Transformation if
(5.19) ⇤T D⇤ = D.
From now on we will use the notation ⇤ to exclusively denote a Lorentz Transfor-
mation. Notice, from (5.19), that any Lorentz transformation satisfies | det(⇤)| = 1.
Any Lorentz Transformation, ⇤, is invertible and (5.19) implies
⇤�1 = D⇤T D.
Therefore, ⇤D⇤T = D, e.g. ⇤T is also a Lorentz Transformation. And ⇤�1 is a
Lorentz Transformation too since
(⇤�1)T D⇤�1 = (D⇤D)D(D⇤T D) = D(⇤D⇤T )D = D3 = D.
164
From (5.19) it su�ces to check
(5.20) ⇤i · ⇤j = Dij, (i, j = 0, 1, 2, 3).
Or equivalently, since ⇤ is Lorentz if and only if ⇤T is Lorentz,
(5.21) ⇤i · ⇤j = Dij, (i, j = 0, 1, 2, 3).
This is just the matrix multiplication (5.19) in component form. A proper or-
thochronous Lorentz transformation satisfies det(⇤) = 1 and ⇤0
0
� 1.
Moreover, by symmetry, (5.19) is equivalent to the following ten equations:
(⇤0
0
)2 ��
(⇤1
0
)2 + (⇤2
0
)2 + (⇤3
0
)2
= 1
(⇤0
1
)2 ��
(⇤1
1
)2 + (⇤2
1
)2 + (⇤3
1
)2
= �1
(⇤0
2
)2 ��
(⇤1
2
)2 + (⇤2
2
)2 + (⇤3
2
)2
= �1
(⇤0
3
)2 ��
(⇤1
3
)2 + (⇤2
3
)2 + (⇤3
3
)2
= �1
⇤0
0
⇤0
1
��
⇤1
0
⇤1
1
+ ⇤2
0
⇤2
1
+ ⇤3
0
⇤3
1
= 0
⇤0
0
⇤0
2
��
⇤1
0
⇤1
2
+ ⇤2
0
⇤2
2
+ ⇤3
0
⇤3
2
= 0
⇤0
0
⇤0
3
��
⇤1
0
⇤1
3
+ ⇤2
0
⇤2
3
+ ⇤3
0
⇤3
3
= 0
⇤0
1
⇤0
2
��
⇤1
1
⇤1
2
+ ⇤2
1
⇤2
2
+ ⇤3
1
⇤3
2
= 0
⇤0
1
⇤0
3
��
⇤1
1
⇤1
3
+ ⇤2
1
⇤2
3
+ ⇤3
1
⇤3
3
= 0
⇤0
2
⇤0
3
��
⇤1
2
⇤1
3
+ ⇤2
2
⇤2
3
+ ⇤3
2
⇤3
3
= 0.
(5.22)
Since ⇤ has sixteen components and is restricted by ten equations, one says that ⇤
has six free parameters.
Perhaps the most well known Lorentz transformation is the boost matrix. Given
v = (v1, v2, v3) 2 R3, we write the boost matrix as
⇤b =
0
B
B
B
B
B
B
@
� ��v1 ��v2 ��v3
��v1 1 + (� � 1)v1v1
|v|2 (� � 1)v2v1
|v|2 (� � 1)v3v1
|v|2
��v2 (� � 1)v1v2
|v|2 1 + (� � 1)v2v2
|v|2 (� � 1)v3v2
|v|2
��v3 (� � 1)v1v3
|v|2 (� � 1)v2v3
|v|2 1 + (� � 1)v3v3
|v|2
1
C
C
C
C
C
C
A
,
165
where � = (1� |v|2)�1/2. Notice that ⇤b has only three free parameters.
In this section, we are exclusively concerned with proper orthochronous Lorentz
transformations ⇤, depending on P and Q, such that
(5.23) ⇤(P + Q) = (p
s, 0, 0, 0)t,
where we recall that s is defined in (5.5).
5.3.1. Lorentz transformations for Relativistic Kinetic Theory. We will
give three examples of Lorentz Transformations satisfying (5.23).
5.3.1.1. Example 1: The Boost Matrix. Our goal is to choose v such that ⇤b
satisfies (5.23). Let
v =p + q
p0
+ q0
, � =p
0
+ q0p
s.
Then ⇤b satisfies (5.23) and is given by
⇤b =
0
B
B
B
B
B
B
@
p0+q0ps
�p1+q1ps
�p2+q2ps
�p3+q3ps
�p3+q3ps
1 + (� � 1) v1v1
|v|2 (� � 1) v2v1
|v|2 (� � 1) v3v1
|v|2
�p3+q3ps
(� � 1) v1v2
|v|2 1 + (� � 1) v2v2
|v|2 (� � 1) v3v2
|v|2
�p3+q3ps
(� � 1) v1v3
|v|2 (� � 1) v2v3
|v|2 1 + (� � 1) v3v3
|v|2
1
C
C
C
C
C
C
A
,
where vivj
|v|2 = (pi
+qi
)(pj
+qj
)
|p+q|2 . By a direct calculation, this example satisfies (5.23).
5.3.1.2. Example 2: To reduce the Collision Integrals. Given p, q 2 R3 with p+q 6=
0, we can always choose three orthonormal vectors
w1, w2, w3 =p + q
|p + q| .
If, for example, p1
+ q1
6= 0 and p2
+ q2
6= 0, then we can explicitly write
w1 =(�(p
2
+ q2
)(p3
+ q3
), 2(p1
+ q1
)(p3
+ q3
),�(p2
+ q2
)(p3
+ q3
))
| (�(p2
+ q2
)(p3
+ q3
), 2(p1
+ q1
)(p3
+ q3
),�(p2
+ q2
)(p3
+ q3
)) |w2 = w1 ⇥ w3/|w1 ⇥ w3|.
If instead p1
+ q1
= 0, then
w1 = (1, 0, 0), w2 = w1 ⇥ w3/|w1 ⇥ w3|.166
With this notation, we can write down a LT which maps p + q ! 0 as
(5.24) ⇤ =
0
B
B
B
B
B
B
@
p0+q0
|P+Q| � p1+q1
|P+Q| � p2+q2
|P+Q| � p3+q3
|P+Q|
0 w1
1
w1
2
w1
3
0 w2
1
w2
2
w2
3
|p+q||P+Q| �p1+q1
|p+q|p0+q0
|P+Q| �p2+q2
|p+q|p0+q0
|P+Q| �p3+q3
|p+q|p0+q0
|P+Q|
1
C
C
C
C
C
C
A
.
This LT clearly satisfies (5.23).
We will derive this LT from the beginning. First, we clearly assume that
⇤1
0
= ⇤2
0
= 0.
Then we are left with the following restrictions for ⇤ to satisfy (5.23):
⇤1
1
(p1
+ q1
) + ⇤1
2
(p2
+ q2
) + ⇤1
3
(p3
+ q3
) = 0
⇤2
1
(p1
+ q1
) + ⇤2
2
(p2
+ q2
) + ⇤2
3
(p3
+ q3
) = 0(5.25)
⇤3
0
(p0
+ q0
) + ⇤3
1
(p1
+ q1
) + ⇤3
2
(p2
+ q2
) + ⇤3
3
(p3
+ q3
) = 0
And the restriction (5.22) reduces to
�
⇤0
0
�
2 � (⇤3
0
)2 = 1
(⇤0
1
)2 ��
(⇤1
1
)2 + (⇤2
1
)2 + (⇤3
1
)2
= �1
(⇤0
2
)2 ��
(⇤1
2
)2 + (⇤2
2
)2 + (⇤3
2
)2
= �1
(⇤0
3
)2 ��
(⇤1
3
)2 + (⇤2
3
)2 + (⇤3
3
)2
= �1
(5.26)
As well as
⇤0
0
⇤0
1
� ⇤3
0
⇤3
1
= 0
⇤0
0
⇤0
2
� ⇤3
0
⇤3
2
= 0
⇤0
0
⇤0
3
� ⇤3
0
⇤3
3
= 0
⇤0
1
⇤0
2
��
⇤1
1
⇤1
2
+ ⇤2
1
⇤2
2
+ ⇤3
1
⇤3
2
= 0
⇤0
1
⇤0
3
��
⇤1
1
⇤1
3
+ ⇤2
1
⇤2
3
+ ⇤3
1
⇤3
3
= 0
⇤0
2
⇤0
3
��
⇤1
2
⇤1
3
+ ⇤2
2
⇤2
3
+ ⇤3
2
⇤3
3
= 0.
(5.27)
167
From (5.26) we choose
(5.28) ⇤0
0
=q
1 + (⇤3
0
)2
Defining
� =⇤3
0
p
1 + (⇤3
0
)2
,
we find that the first three equations in (5.27) are
(5.29) ⇤0
1
= �⇤3
1
, ⇤0
2
= �⇤3
2
, ⇤0
3
= �⇤3
3
.
And we plug (5.29) into the bottom three equations in (5.27) to find
⇤1
1
⇤1
2
+ ⇤2
1
⇤2
2
+ ⇤3
1
⇤3
2
(1� �2) = 0
⇤1
1
⇤1
3
+ ⇤2
1
⇤2
3
+ ⇤3
1
⇤3
3
(1� �2) = 0
⇤1
2
⇤1
3
+ ⇤2
2
⇤2
3
+ ⇤3
2
⇤3
3
(1� �2) = 0.
(5.30)
We further plug (5.29) into the last three equations in (5.26) to obtain
(⇤1
1
)2 + (⇤2
1
)2 + (⇤3
1
)2(1� �2) = 1
(⇤1
2
)2 + (⇤2
2
)2 + (⇤3
2
)2(1� �2) = 1
(⇤1
3
)2 + (⇤2
3
)2 + (⇤3
3
)2(1� �2) = 1
(5.31)
We have thus reduced the system to nine equations and nine unknowns; it remains
to simultaneously satisfy (5.25), (5.30) and (5.31).
We first concentrate on satisfying (5.25). In any case, with these vectors define
⇤1 = (0, w1),
⇤2 = (0, w2),
⇤3 =�
⇤3
0
,�w3(1� �2)�1/2
�
=
✓
⇤3
0
,�w3
q
1 + (⇤3
0
)2
◆
.
(5.32)
These choices clearly satisfy (5.25) as soon as
⇤3
0
(p0
+ q0
) = |p + q|q
1 + (⇤3
0
)2.
168
Squaring and choosing the positive root by necessity, we have
⇤3
0
=|p + q||P + Q| , ⇤0
0
=p
0
+ q0
|P + Q| .
It remains to show that these choices satisfy (5.30) and (5.31).
To this end, we define the 3⇥ 3 matrix
U =
0
B
B
B
@
⇤1
1
⇤1
2
⇤1
3
⇤2
1
⇤2
2
⇤2
3
⇤3
1
p
1� �2 ⇤3
2
p
1� �2 ⇤3
3
p
1� �2
1
C
C
C
A
where the coe�cients above are defined by (5.32). By definition, UUT = I3
. Therefore
U is invertible and
UT U = UT (UUT )U = UT (UUT )�1U = UT (UT )�1U�1U = I3
,
which means UT = U�1. This last display also says that the parameters (5.32) satisfy
(5.30) and (5.31).
5.3.1.3. Example 3: Hilbert-Schmidt form for the Linearized Operator. We now
wish to derive a Lorentz Transformation which satisfies (5.23) but also
(5.33) ⇤(P �Q) = (0, 0, 0, g).
In [15, p.277] and Section 5.6 this transformation is used to write down a Hilbert-
Schmidt form for the linearized collision operator. But the transformation is not
written down explicitly anywhere. All that is given is the conditions (5.23), (5.33)
and the first row ⇤0
. However, from this information, we can make the following
educated guess for ⇤ satisfying (5.23) and (5.33):
⇤ =
0
B
B
B
B
B
B
@
p0+q0ps
�p1+q1ps
�p2+q2ps
�p3+q3ps
⇤1
0
⇤1
1
⇤1
2
⇤1
3
0 (p⇥q)1|p⇥q|
(p⇥q)2|p⇥q|
(p⇥q)3|p⇥q|
p0�q0
g�p1�q1
g�p2�q2
g�p3�q3
g
1
C
C
C
C
C
C
A
.
169
Clearly, ⇤i · ⇤j = Dij if i, j 2 {0, 2, 3}. By the Lorentz condition ⇤0
· ⇤0
= 1, we
choose ⇤1
0
as
(5.34) (⇤1
0
)2 =
✓
p0
+ q0p
s
◆
2
�✓
p0
� q0
g
◆
2
� 1.
By further computations we have
sg2(⇤1
0
)2 = g2 (p0
+ q0
)2 � s (p0
� q0
)2 � sg2
= 2(P ·Q� c2) (p0
+ q0
)2 � 2(P ·Q + c2) (p0
� q0
)2 � 4((P ·Q)2 � c4)
= 2(P ·Q� c2)�
p2
0
+ q2
0
+ 2p0
q0
�
� 2(P ·Q + c2)�
p2
0
+ q2
0
� 2p0
q0
�
�4((P ·Q)2 � c4)
= 8(P ·Q)p0
q0
� 4c2(p2
0
+ q2
0
)� 4((P ·Q)2 � c4)
= 8(P ·Q)p0
q0
� 4c2(p2
0
+ q2
0
)� 4((p0
q0
)2 + (p · q)2 � 2p0
q0
(p · q)� c4)
= 4p2
0
q2
0
� 4(p · q)2 � 4c2(p2
0
+ q2
0
) + 4c4
= 4�
|p|2|q|2 � (p · q)2
�
= 4|p⇥ q|2.
Choosing the positive root, we conclude
⇤1
0
=2|p⇥ q|p
sg=
|p⇥ q|p
(P ·Q)2 � c4
.
The coe�cients ⇤1
i (i = 1, 2, 3) are similarly completely determined by the Lorentz
condition ⇤i ·⇤i = �1. However, this turns out to be a long computation. In the end
we obtain
⇤1
i =2 (pi {p0
� q0
P ·Q}+ qi {q0
� p0
P ·Q})psg|p⇥ q| (i = 1, 2, 3).
Now, we have a complete description of this lorentz transformation in terms of p, q.
The only way to derive this matrix was by guessing the first, third and fourth
rows and then computing ⇤i · ⇤i = Dii. However, it turns out that it is much easier
to verify that this is indeed a Lorentz Transformation by checking ⇤i · ⇤j = Dij
(i, j = 0, 1, 2, 3), e.g. the conditions (5.21). We will do this one row at a time.
170
The First row. Clearly ⇤0 · ⇤j = D0j for j = 0, 2, 3. To check ⇤0 · ⇤1 = D
01
= 0
we compute
3
X
i=1
⇤1
i pi =2p
sg|p⇥ q|�
|p|2�
c2p0
� q0
P ·Q
+ p · q�
c2q0
� p0
P ·Q �
=2 (|p|2c2p
0
� |p|2p0
q2
0
+ |p|2q0
p · q + p · qc2q0
� p · qp2
0
q0
+ p0
(p · q)2)psg|p⇥ q|
=2 (�p
0
|p|2|q|2 + p0
(p · q)2)psg|p⇥ q| =
2psg|p⇥ q|
�
�|p⇥ q|2p0
�
(5.35)
= �2|p⇥ q|psg
p0
.
And similarly,
X
i
⇤1
i qi =2p
sg|p⇥ q|�
p · q�
c2p0
� q0
P ·Q
+ |q|2�
c2q0
� p0
P ·Q �
=2 (p · qc2p
0
� p · qp0
q2
0
+ q0
(p · q)2 + |q|2c2q0
� |q|2p2
0
q0
+ |q|2p0
p · q)psg|p⇥ q|(5.36)
=2 (�|q|2|p|2q
0
+ q0
(p · q)2)psg|p⇥ q| = �2|p⇥ q|p
sgq0
.
We thus conclude that ⇤0 · ⇤1 = D01
= 0.
The Second row. The calculation we will check now is ⇤1 · ⇤1 = D11
= �1. The
rest of the conditions are checked elsewhere. We compute
sg2|p⇥ q|24
(⇤1
i )2 =
�
pi
�
c2p0
� q0
P ·Q
+ qi
�
c2q0
� p0
P ·Q �
2
=⇣
p2
i
�
c2p0
� q0
P ·Q
2
+ q2
i
�
c2q0
� p0
P ·Q
2
⌘
+�
2piqi
�
c2p0
� q0
P ·Q �
c2q0
� p0
P ·Q �
.
Summing
sg2|p⇥ q|24
3
X
i=1
(⇤1
i )2 =
⇣
|p|2�
c2p0
� q0
P ·Q
2
+ |q|2�
c2q0
� p0
P ·Q
2
⌘
+�
2p · q�
c2p0
� q0
P ·Q �
c2q0
� p0
P ·Q �
.
171
Notice that the r.h.s. above is
=�
|p|2�
c2p0
� q0
P ·Q
+ p · q�
c2q0
� p0
P ·Q � �
c2p0
� q0
P ·Q
+�
|q|2�
c2q0
� p0
P ·Q
+ p · q�
c2p0
� q0
P ·Q � �
c2q0
� p0
P ·Q
.
Therefore, by (5.35) and (5.36), we conclude that
3
X
i=1
(⇤1
i )2 = � 4|p⇥ q|2
sg2|p⇥ q|2�
p0
�
c2p0
� q0
P ·Q
+ q0
�
c2q0
� p0
P ·Q �
= � 4
sg2
�
p0
�
c2p0
� q0
P ·Q
+ q0
�
c2q0
� p0
P ·Q �
=1
sg2
�
�4c2(p2
0
+ q2
0
) + 8p0
q0
(P ·Q)�
.
Adding and subtracting terms, we obtain
=1
sg2
�
�4c2(p2
0
+ q2
0
) + (2(p2
0
+ q2
0
)� 2(p2
0
+ q2
0
) + 8p0
q0
)P ·Q
=1
sg2
�
�4c2(p2
0
+ q2
0
) + 2((p0
+ q0
)2 � 2(p0
� q0
)2)P ·Q
Adding and subtracting terms again, we obtain
=1
sg2
�
�4c2(p2
0
+ q2
0
+ p0
q0
� p0
q0
) + 2((p0
+ q0
)2 � 2(p0
� q0
)2)P ·Q
=1
sg2
�
�2c2((p0
+ q0
)2 + (p0
� q0
)2) + 2((p0
+ q0
)2 � 2(p0
� q0
)2)P ·Q
=2(P ·Q� c2)
sg2
(p0
+ q0
)2 � 2(P ·Q + c2)
sg2
(p0
� q0
)2 .
And by (5.7) this says
3
X
i=1
(⇤1
i )2 =
✓
p0
+ q0p
s
◆
2
�✓
p0
� q0
g
◆
2
.
Therefore, by (5.34), ⇤1 · ⇤1 = D11
= �1.
The Third row. By the definition of the cross product, we clearly have
⇤2 · ⇤j = D2j,
for j = 0, 1, 2, 3.
The Fourth row. This is similar to the first row. Notice that ⇤3 · ⇤j = D3j for
j = 0, 2, 3. And again by (5.35) and (5.36) we conclude that ⇤3 · ⇤1 = D31
172
We remark that all the components of this Lorentz Transformation may have been
previously unknown in their precise form1.
5.4. Center of Momentum Reduction of the Collision Integrals
In this section, we will reduce the collision integrals in the center of momentum
system. Given (5.3) and a integrable function G : R3 ⇥ R3 ⇥ R3 ⇥ R3 ! R we have
c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
s�(4)(P 0 + Q0 � P �Q)G(P, Q, P 0, Q0)
=
Z
R3
dq
Z
S2
d!v�(g, ✓)G(P, Q, P 0, Q0)
where v is given by (5.12) and the angle ✓ is given by
(5.37) cos ✓ =p
|p| · !,
where p is defined by ⇤(P �Q) = (0, p)t. The post-collisional velocities satisfy
(5.38) P 0 =1
2⇤�1
0
@
s
g!
1
A , Q0 =1
2⇤�1
0
@
s
�g!
1
A ,
where ⇤ is any Lorentz Transformation which satisfies (5.23). In particular any of
the Lorentz Transformations in the previous section will do.
We will compute, from (5.2), explicitlyZ
R3
dq0
q00
Z
R3
dp0
p00
s�(g, ✓)�(4)(P + Q� P 0 �Q0)G(P, Q, P 0, Q0),
=
Z
S2
d!gp
s
2�(g, ✓)G(P, Q, P 0, Q0).(5.39)
This gives the reduced expression by (5.12).
We first claim that the r.h.s. of (5.39) is
=
Z
R3
dq0
q00
Z
R3
dp0
p00
s�(g, ✓)�(4)(⇤(P + Q)� P 0 �Q0)G(P, Q,⇤�1P 0,⇤�1Q0).
This holds because dq0
q00and dp0
p00are Lorentz invariant measures and
�(4)(⇤(P + Q� P 0 �Q0)) = �(4)(P + Q� P 0 �Q0).
1W. A. van Leeuwen, personal communication, 2003
173
But notice that the claim is not true unless the angle ✓ is redefined as
cos ✓ =[⇤(P �Q)] · (P 0 �Q0)
|P �Q|2 =p
|p| ·(p0 � q0)|P �Q| ,
where we have used |P �Q| = |⇤(P �Q)| = |p|.
Since, now, p0+ q0 = 0, the integration over q0 can be carried out immediately and
we obtain
=1
2
Z
dp0
(p00
)2
s�(g, ✓)�(1
2
ps� p0
0
)G(P, Q,⇤�1P 0,�⇤�1P 0).
where now P 0 = (�p00
, p0) and, using p0 + q0 = 0, the angle is
cos(✓) = 2p
|p| ·p0
|P �Q|
Next change to polar coordinates as p0 = |p0|! (! 2 S2) and
dp0 = |p0|2d|p0|d!.
We use the following calculation to compute the delta function
1
2
ps� p0
0
=1
2
ps�
p
c2 + |p0|2 =1
4
s� (c2 + |p0|2)1
2
ps +
p
c2 + |p0|2
=1
4
g2 � |p0|21
2
ps +
p
1 + |p0|2.
We have just used (5.8). Thus
1
2
ps� p0
0
=(1
2
g � |p0|)(1
2
g + |p0|)1
2
ps +
p
c2 + |p0|2.
Note that if 1
2
g = |p0| then (5.8) grantsp
c2 + |p0|2 = 1
2
ps. Therefore,
�(1
2
ps� p0
0
) =1
2
ps +
p
c2 + |p0|21
2
g + |p0| �(1
2g � |p0|) =
ps
g�(
1
2g � |p0|).
We plug this in to obtain
=1
2
Z 1
0
d|p0|Z
S2
d!|p0|2(p0
0
)2
s3/2
g�(g, ✓)�(
1
2g � |p0|)F (⇤�1!)G(⇤�1!).
where
! = (p00
, |p0|!)T =1
2(p
s, g!)t, ! = (p00
,�|p0|!)T =1
2(p
s,�g!)t.
174
Further, since p0 = |p0|! = 1
2
g!, the angle ✓ is redefined (from above) as (5.37). We
now evaluate the delta function to obtain (5.39).
5.4.1. Post-Collisional velocities. Notice that, using these definition (5.38)
for the post-collisional velocities, we recover the identities of conservation for elastic
collisions (5.4). This is verified by multiplying both sides of (5.4) by ⇤ since
⇤(P + Q) = (p
s, 0, 0, 0)t = ! + ! = ⇤(P 0 + Q0).
Since ⇤�1 is also a Lorentz Transformation, we have
P 0 · P 0 = Q0 ·Q0 = c2,
which is also as it should be...
We further assume
⇤0
0
=p
0
+ q0
|P + Q| , ⇤0
i = � pi + qi
|P + Q| (i = 1, 2, 3).
Note that this assumption is satisfied by all the examples in this note. Define
⇤i = (⇤1
i ,⇤2
i ,⇤3
i ) (i = 1, 2, 3).
We can then write
p0i =pi + qi
2+ g
! · ⇤i
2,
q0i =pi + qi
2� g
! · ⇤i
2.
If ⇤ is the boost matrix, we have exactly (5.13).
5.5. Glassey-Strauss Reduction of the Collision Integrals
Glassey & Strauss have derived [32] an alternative form for collision operator
without using the center-of-momentum system in (5.14) with kernel (5.15) and post-
collisional momentum (5.16). We will write down their argument as follows.
175
Given an integrable function G : R3 ⇥ R3 ⇥ R3 ⇥ R3 ! R, we will show
c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
�(4)(P 0 + Q0 � P �Q)G(P, Q, P 0, Q0)
=1
p0
Z
R3
dq
q0
Z
S2
d!B(p, q,!)G(P, Q, P 0, Q0),
where B(p, q,!) is given by (5.15) and (P 0, Q0) on the r.h.s. are given by (5.16).
We focus on
I ⌘Z
R3
dq0
q00
Z
R3
dp0
p00
�(4)(P 0 + Q0 � P �Q)G(p, q, p0, q0).
It is su�cient to establish
I =2
c
Z
S2
d!B(p, q,!)G(p, q, p0, q0),
where (p0, q0) are given by (5.16). We split I = 1
2
I + 1
2
I.
Letting q0 = p + q � p0 we can immediately remove three of the delta functions.
Next translate p0 ! p + p0 so that q0 ! q � p0. And then switch to polar coordinates
as p0 = r!, dp0 = r2drd! where r 2 [0,1) and ! 2 S2. Then we have
1
2I =
1
2
Z 1
0
Z
S2
r2drd!
p00
q00
�(p00
+ q00
� p0
� q0
)G(p, q, p0, q0),
where p0 = p + r! and q0 = q � r!.
Now consider the other half 1
2
I. Let p0 = p + q � q0 and remove three of the
delta functions. Next translate q0 ! q + q0 so that p0 ! p� q0. And switch to polar
coordinates as q0 = r!, dq0 = r2drd! where r 2 [0,1) and ! 2 S2. Then
1
2I =
1
2
Z 1
0
Z
S2
r2drd!
p00
q00
�(p00
+ q00
� p0
� q0
)G(p, q, p0, q0),
where p0 = p� r! and q0 = q + r!. Further change variables r ! �r so that
1
2I =
1
2
Z
0
�1
Z
S2
r2drd!
p00
q00
�(p00
+ q00
� p0
� q0
)G(p, q, p0, q0),
where now p0 = p + r! and q0 = q � r!.
We combine the last two splittings to conclude that
I =1
2I +
1
2I =
1
2
Z
+1
�1
Z
S2
r2drd!
p00
q00
�(p00
+ q00
� p0
� q0
)G(p, q, p0, q0),
where p0 = p + r! and q0 = q � r!.
176
We now focus on the argument of the delta function. For �i > 0 (i = 1, 2), we use
the identity �(�1
� �2
) = 2�1
�(�2
1
� �2
2
) to see that
�(p00
+ q00
� p0
� q0
) = 2(p0
+ q0
)�((p00
+ q00
)2 � (p0
+ q0
)2)
= 2(p0
+ q0
)�(2p00
q00
� {(p0
+ q0
)2 � (p00
)2 � (q00
)2}).
If {(p0
+ q0
)2 � (p00
)2 � (q00
)2} > 0 then this is
= 8(p0
+ q0
)p00
q00
�(4(p00
)2(q00
)2 � {(p0
+ q0
)2 � (p00
)2 � (q00
)2}2).
If {(p0
+ q0
)2 � (p00
)2 � (q00
)2} < 0, then the delta function is zero. Now write the
argument of the delta function as
p(r) = �(p0
+ q0
)4 + 2(p0
+ q0
)2{(p00
)2 + (q00
)2}� {(p00
)2 + (q00
)2}2 + 4(p00
)2(q00
)2
= �(p0
+ q0
)4 + 2(p0
+ q0
)2{(p00
)2 + (q00
)2}� {(p00
)2 � (q00
)2}2.
Plugging in p0 = p + r! and q0 = q � r! we observe that
(p00
)2 � (q00
)2 = |p + r!|2 � |q � r!|2 = p2
0
� q2
0
+ 2r! · (p + q).
This means that p(r) is quadratic in r. Moreover,
p(0) = �(p0
+ q0
)4 + 2(p0
+ q0
)2{(p0
)2 + (q0
)2}� {(p0
)2 � (q0
)2}2
= �(p0
+ q0
)4 + 2(p0
+ q0
)2{(p0
)2 + (q0
)2}� (p0
+ q0
)2(p0
� q0
)2
= �(p0
+ q0
)4 + (p0
+ q0
)2{(p0
)2 + (q0
)2}+ 2p0
q0
(p0
+ q0
)2 = 0.
We conclude that p(r) = 4D1
r2 � 8D2
r for some D1
, D2
2 R.
We will now determine D1
, D2
. Expanding
(p00
)2 + (q00
)2 = 2c2 + |p + r!|2 + |q � r!|2 = p2
0
+ q2
0
+ 2r! · (p� q) + 2r2.
We plug the last few calculations into p(r) to write it out in terms of r as
p(r) = �(p0
+ q0
)4 + 2(p0
+ q0
)2{p2
0
+ q2
0
+ 2r! · (p� q) + 2r2}
� {p2
0
� q2
0
+ 2r! · (p + q)}2
= �(p0
+ q0
)4 + 2(p0
+ q0
)2{p2
0
+ q2
0
+ 2r! · (p� q) + 2r2}
� {p2
0
� q2
0
}2 � 4r2{! · (p + q)}2 � 4r{! · (p + q)}{p2
0
� q2
0
}.177
Rearrainging the terms
p(r) = 4{(p0
+ q0
)2 � {! · (p + q)}2}r2
+ 4�
(p0
+ q0
)2{! · (p� q)}� {! · (p + q)}{p2
0
� q2
0
}
r
� (p0
+ q0
)4 + 2(p0
+ q0
)2{p2
0
+ q2
0
}� {p2
0
� q2
0
}2
= 4{(p0
+ q0
)2 � {! · (p + q)}2}r2
+ 4�
(p0
+ q0
)2{! · (p� q)}� {! · (p + q)}{p2
0
� q2
0
}
r.
In other words,
D1
= {(p0
+ q0
)2 � {! · (p + q)}2},
2D2
= �(p0
+ q0
)2{! · (p� q)}+ {! · (p + q)}{p2
0
� q2
0
}.
We further calculate D2
as
2D2
= �(p2
0
+ q2
0
+ 2p0
q0
){! · (p� q)}+ {! · (p + q)}{p2
0
� q2
0
}
= w · p�
�2q2
0
� 2p0
q0
+ ! · q�
2p2
0
+ 2p0
q0
= 2{(p0
+ q0
)! · (p0
q � q0
p)}
= 2(p0
+ q0
)p0
q0
⇢
! ·✓
q
q0
� p
p0
◆�
.
Thus, p(r) = 4D1
r2 � 8D2
r with these definitions.
Plug this formulation for p(r) into the full integral to obtain
I =1
2
Z
+1
�1
Z
S2
r2drd!8(p0
+ q0
)�(4D1
r2 � 8D2
r)G(p, q, p0, q0).
Equivalently
I =
Z
+1
�1
Z
S2
r2drd!(p
0
+ q0
)
D1
�(r{r � 2D2
/D1
})G(p, q, p0, q0).
We use the identity �(r{r�2D2
/D1
}) =�
�
�
D12D2
�
�
�
{�(r)+�(r�2D2
/D1
)}. The first delta
function drops out because of the r2 factor. We thus have
I =
Z
+1
�1
Z
S2
r2drd!(p
0
+ q0
)
2|D2
| �(r � 2D2
/D1
)G(p, q, p0, q0),
where we have used D1
� 2 [31]; but D2
can be negative. Evaluating the delta
function we obtain the result.
178
5.6. Hilbert-Schmidt form
We linearize the collision operator around a relativistic Maxwellian, J , as
F = J(1 + f),
where the relativistic Maxwellian is defined in (5.10). Since C(J, J) ⌘ 0, the linearized
collision operator takes the form
Lf ⌘ �J�1(p) {C(Jf, J) + C(J, Jf)} .
Define the collision frequency by
⌫(p) ⌘ c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
W (p, q|p0, q0)J(q).
Then we can write
Lf = ⌫(p)f �Kf,
where K = K2
�K1
and
K1
f ⌘ c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
W (p, q|p0, q0)J(q)f(q),
K2
f ⌘ c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
W (p, q|p0, q0)J(q) {f(p0) + f(q0)} .
We define
k1
(p, q) =c
2
J(q)
p0
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
W (p, q|p0, q0),
which can be simplified as in either Section 5.4 or Section 5.5.
Then we have a Hilbert-Schmidt form for K1
. Our goal in this section is to derive
a Hilbert-Schmidt form for K2
. From (5.3) we have
K2
f =c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
s�(g, ✓)�(4)(P + Q� P 0 �Q0)J(q) {f(p0) + f(q0)} .
We first reduce this to a Hilbert-Schmidt form and second carry out the delta function
integrations in the kernel. This calculation is from [15, p.277], see also [22].
In preparation, we write down some invariant quantities. By (5.4) and (5.5)
(P �Q) · (P 0 �Q0) = 2P · P 0 + 2Q ·Q0 � (P + Q) · (P 0 + Q0)
= 2P · P 0 + 2Q ·Q0 � s.
179
Further notice that (5.4) implies
(P � P 0) · (P � P 0) = (Q0 �Q) · (Q0 �Q).
Expanding this we have
2c2 � P · P 0 = 2c2 �Q0 ·Q.
Thus P ·P 0 = Q ·Q0. Let g = g(P, P 0) from (5.7). We will always use g to exclusively
denote g = g(P, Q). By (5.8), we redefine ✓ from (5.9) as
cos ✓ = �(P �Q) · (P 0 �Q0)/g2 = 1� 2
✓
g
g
◆
2
.
See [30, p.111-113] for lots of similar calculations. We further claim that
(5.40) g2 = g2 +1
2(P + P 0) · (Q + Q0 � P � P 0),
Let s = s(P, P 0) = g2 + 4c2. Then (5.40) is equivalent to
g2 =1
2g2 � 2c2 +
1
2(P + P 0) · (Q + Q0),
=1
2g2 + g2 � 2P ·Q +
1
2(P + P 0) · (Q + Q0).
We thus prove (5.40) by showing that
1
2g2 � 2P ·Q +
1
2(P + P 0) · (Q + Q0) = 0.
Expanding the r.h.s we obtain
P · P 0 � c2 � 2P ·Q +1
2P ·Q +
1
2P ·Q0 +
1
2P 0 ·Q +
1
2P 0 ·Q0.
Since P ·Q = P 0 ·Q0 and P · P 0 = Q ·Q0 because of (5.4), we obtain
�P ·Q +1
2P · P 0 +
1
2Q ·Q0 � c2 +
1
2P ·Q0 +
1
2P 0 ·Q,
which by (5.4) is
�P ·Q� c2 +1
2(P + Q) · (P 0 + Q0) = �P ·Q� c2 +
1
2s = 0.
This establishes the claim (5.40).
180
Now we establish the Hilbert-Schmidt form. First consider
c
2
1
p0
Z
R3
dq
q0
Z
R3
dq0
q00
Z
R3
dp0
p00
s�(g, ✓)�(4)(P + Q� P 0 �Q0)J(q)f(p0).
Exchanging q with p0 the integral above is equal to
c
2
1
p0
Z
dq
q0
f(q)
⇢
Z
dq0
q00
Z
dp0
p00
s�(g, ✓)�(4)(P + P 0 �Q�Q0)J(p0)�
,
where ✓ is now
(5.41) cos ✓ = 1� 2
✓
g
g
◆
2
,
and from (5.40)
(5.42) g2 = g2 +1
2(P + Q) · (P 0 + Q0 � P �Q),
and s is defined by
(5.43) s = g2 + 4c2.
We do a similar calculation for the second term in K2
f , e.g. exchange q with q0 and
then swap the q0 and p0 notation. Therefore, we can define
(5.44) k2
(p, q) ⌘ c
p0
q0
Z
dq0
q00
Z
dp0
p00
s�(g, ✓)�(4)(P + P 0 �Q�Q0)J(p0).
And we can now write the Hilbert-Schmidt form K2
f =R
k2
(p, q)f(q)dq.
We will carry out the delta function integrations in k2
(p, q) using a center-of-
momentum system. One might try, instead of using the center-of-momentum system,
to carry out the delta function integrations using a similar procedure to the Appendix
in [32], which we did in Section 5.5.
Next, we want to translate (5.44) into an expression involving the total and relative
momentum variables, P 0 + Q0 and P 0�Q0 respectively. Define u by u(r) = 0 if r < 0
and u(r) = 1 if r � 0. Let g = g(P 0, Q0) and s = s(P 0, Q0) . We claim that
(5.45)
Z
R3
dq0
q00
Z
R3
dp0
p00
G(P, Q, P 0, Q0) =1
16
Z
R4⇥R4
dµ(P 0, Q0)G(P, Q, P 0, Q0),
where we are now integrating over (P 0, Q0) and
dµ(P 0, Q0) = dP 0dQ0u(p00
+ q00
)u(s� 4c2)�(s� g2 � 4c2)�((P 0 + Q0) · (P 0 �Q0)).
181
To establish the claim, first notice that
(P 0 + Q0) · (P 0 �Q0) = P 0 · P 0 �Q0 ·Q0
= (p00
)2 � |p0|2 � (q00
)2 + |q0|2 = Ap � Aq,
where now p00
and q00
are integration variables and we have defined
Ap = (p00
)2 � [|p0|2 + c2], Aq = (q00
)2 � [|q0|2 + c2].
Integrating first over dP 0, see that alternatively
(P 0 + Q0) · (P 0 �Q0) = (p00
)2 � [|p0|2 + c2 + Aq]
=
⇢
p00
�q
|p0|2 + c2 + Aq
�⇢
p00
+q
|p0|2 + c2 + Aq
�
.
Furthermore, by (5.5) and (5.7) we have
s� g2 � 4c2 = (P 0 + Q0) · (P 0 + Q0) + (P 0 �Q0) · (P 0 �Q0)� 4c2
= 2P 0 · P 0 + 2Q0 ·Q0 � 4c2
= 2Ap + 2Aq,
Thus, similarly
s� g2 � 4c2 = 2(p00
)2 � 2[|p0|2 + c2 � Aq]
=
⇢
p00
�q
|p0|2 + c2 � Aq
�⇢
p00
+q
|p0|2 + c2 � Aq
�
.
Further note that p00
+ q00
� 0 and s�4c2 � 0 together imply p00
� 0 and q00
� 0. With
these expressions, by standard delta function calculations we establish (5.45).
We thus conclude that
k2
(p, q) ⌘ c
p0
q0
1
16
Z
R4⇥R4
dµ(P 0, Q0)s�(g, ✓)�(4)(P + P 0 �Q�Q0)J(p0).
Now apply the change of variables
P = P 0 + Q0, Q = P 0 �Q0.
This transformation has Jacobian = 16 and inverse tranformation as
P 0 =1
2P +
1
2Q, Q0 =
1
2P � 1
2Q.
182
With this change of variable the integral becomes
k2
(p, q) ⌘ 1
4⇡K2
(c2)p0
q0
Z
R4⇥R4
dµ(P , Q)s�(g, ✓)�(4)(P �Q + Q)e�c
2 (p0+q0),
where the measure is
dµ(P , Q) = dPdQu(p0
)u(P · P � 4c2)�(P · P + Q · Q� 4c2)�(P · Q).
Also g � 0 from (5.42) is now given by
g2 = g2 +1
2(P + Q) · (P � P �Q).
And ✓ and s can be defined through the new g as in (5.41) and (5.43).
We next carry out the delta function argument for �(4)(P �Q + Q) to obtain
k2
(p, q) ⌘ 1
4⇡K2
(c2)p0
q0
Z
R4
dµ(P )s�(g, ✓)e�c
2 (p0+q0�p0),
where the measure is
dµ(P ) = dPu(p0
)u(P · P � 4c2)�(P · P � g2 � 4c2)�(P · (Q� P )).
By (5.8) we have
u(p0
)�(P · P � g2 � 4c2) = u(p0
)�(P · P � s)
= u(p0
)�((p0
)2 � |p|2 � s)
=�(p
0
�p
|p|2 + s)
2p
|p|2 + s.
We then carry out one integration using delta function to get
k2
(p, q) ⌘ e�c
2 (q0�p0)
8⇡K2
(c2)p0
q0
Z
R3
dp
p0
u(P · P � 4c2)�(P · (Q� P ))s�(g, ✓)e�c
2 p0 ,
with p0
⌘p
|p|2 + s. By (5.8), we now have
P · P � 4c2 = s� 4c2 = g2 � 0.
So always u(P · P � 4c2) = 1 and integral reduces to
k2
(p, q) ⌘ e�c
2 (q0�p0)
8⇡K2
(c2)p0
q0
Z
R3
dp
p0
�(P · (Q� P ))s�(g, ✓)e�c
2¯P · ¯U ,
where U = (1, 0, 0, 0)t and e�c
2 p0 = e�c
2¯P · ¯U .
183
We finish our reduction by moving to a center-of-momentum system (5.6). As this
form suggests, we choose the Lorentz Transformation ⇤ from Section 5.3.1.3 which
satisfies (5.23) and (5.33). Define U = ⇤U , then Definition 5.1 gives
Z
dp
p0
s�(g, ✓)e�c
2 p0�(P · (Q� P )) =
Z
dp
p0
s⇤
�(g⇤
, ✓⇤
)e�c
2¯P ·U�(P · ⇤(Q� P )).
where g⇤
, s⇤
� 0 are now given by
g2
⇤
= g2 +1
2⇤(P + Q) · {P � ⇤(P + Q)} = g2 +
1
2
ps{p
0
�p
s}
s⇤
= 4c2 + g2
⇤
cos ✓⇤
= 1� 2
✓
g
g⇤
◆
2
.
(5.46)
The equality of the two integrals holds because dp/p0
is a Lorentz invariant measure.
We now work with the integral on the left hand side above. In this coordinate
system
P · ⇤(Q� P ) = �p3
g.
We switch to polar coordinates in the form
dp = |p|2d|p| sin d d', p ⌘ |p|(sin cos', sin sin', cos ).
Then we can write k2
(p, q) as
e�c
2 (q0�p0)
8⇡K2
(c2)p0
q0
Z
2⇡
0
d'
Z ⇡
0
sin d
Z 1
0
|p|2d|p|p
0
s⇤
�(g⇤
, ✓⇤
)e�c
2¯P ·U�(|p|g cos ).
From Section 5.3.1.3 we have
U = ⇤U =
✓
p0
+ q0p
s,2|p⇥ q|
gp
s, 0,
p0
� q0
g
◆t
.
We evaluate the last delta function at = ⇡/2 to write k2
(p, q) as
(5.47)e�
c
2 (q0�p0)
8⇡gK2
(c2)p0
q0
Z
2⇡
0
d'
Z 1
0
|p|d|p|p
0
s⇤
�(g⇤
, ✓⇤
)e�c
2 p0p0+q0p
s ec|p⇥q|g
ps
|p| cos '.
This is already a useful reduced form for k2
(p, q).
184
However, we now further convert the integration over d|p| to an integration over
d⇥ ⌘ d✓⇤
. First notice that (5.46) implies
(5.48) g = g⇤
r
1� cos ✓⇤
2= g
⇤
sin✓⇤
2.
Now (5.46) implies g2
�
sin�2
✓⇤2
� 1�
= 1
2
ps(p
0
�ps). Equivalently,
(5.49) 2g2 cot2
✓⇤
2=p
s(p0
�p
s).
Thus 2 g2psd�
cot2
⇥
2
�
= dp
|p|2 + s, or
g2
ps
d cos⇥
sin4 ⇥
2
=|p|d|p|
p0
.
This last deduction follows since
d
✓
cot2
⇥
2
◆
= d
cos2
⇥
2
sin2 ⇥
2
!
= �cos ⇥
2
sin ⇥
2
sin2 ⇥
2
d⇥�cos3
⇥
2
sin3 ⇥
2
d⇥
= � cos⇥
2
sin2 ⇥
2
+ cos2
⇥
2
sin3 ⇥
2
!
d⇥
= �1
2
sin⇥d⇥
sin4 ⇥
2
=1
2
d cos⇥
sin4 ⇥
2
,
where we used the trigonometric formula cos ⇥
2
sin ⇥
2
= 1
2
sin⇥ in the last step. Al-
ternatively, we can solve (5.49) for the change of variable
(5.50) |p| = 2gs�1/2
4c2 +g2
sin2 ✓2
!
1/2
cot⇥
2.
Here 0 |p| 1 and, therefore, 0 ⇥ ⇡ but the ⇥ goes from ⇡ back to 0 as
|p| goes from 0 to 1. We plug this change of variable into the integral (5.47) using
(5.46) and (5.48) to establish
k2
(p, q) =e�
c
2 (q0�p0)
8⇡K2
(c2)p0
q0
gps
Z
2⇡
0
d'
Z ⇡
0
sin⇥d⇥
sin4 ⇥
2
4c2 +g2
sin2 ⇥
2
!
�
g
sin ⇥
2
,⇥
!
⇥ exp
✓
� c
2
p0
+ q0p
sp
0
+ c|p⇥ q|gp
s|p| cos'
◆
,(5.51)
where p0
=p
s + |p|2 and |p| defined by (5.50).
185
We can reduce (5.51) even further and evaluate all the integrals if we assume
� = 1. We write k2
(p, q) from (5.47), and using (5.46), as
ec
2 (q0�p0)
16⇡gK2
(c2)p0
q0
Z 1
0
ydyp
s + y2
⇣
s +p
s2 + sy2
⌘
e�c
2p0+q0p
s
ps+y2
I0
✓
c|p⇥ q|gp
sy
◆
.
where I0
(y) = 1
2⇡
R
2⇡
0
ey cos 'd' is a modified Bessel function of index zero. By the
change of variable y ! yp
s we can write k2
(p, q) as
s3/2ec
2 (q0�p0)
16⇡gK2
(c2)p0
q0
Z 1
0
ydyp
1 + y2
⇣
1 +p
1 + y2
⌘
e�c
2 (p0+q0)
p1+y2
I0
✓
c|p⇥ q|
gy
◆
,
This is a Laplace Transform and a known integral, which can be calculated exactly
via a taylor expansion [46, p.134]. For instance, [32, Lemma 3.5 on p.322] says that
for any R > r � 0 we have
Z 1
0
e�Rp
1+y2yI
0
(ry)p
1 + y2
dy =e�p
R2�r2
pR2 � r2
,
Z 1
0
e�Rp
1+y2yI
0
(ry)dy =R
R2 � r2
⇢
1 +1p
R2 � r2
�
e�p
R2�r2.
See also [58] or [57, p.383]. Using these formula’s we can express the integral as
k2
(p, q) =s3/2e
c
2 (q0�p0)
16⇡gK2
(c2)p0
q0
H1
(p, q) exp (�H2
(p, q)) ,
where H2
(p, q) =p
{c(p0
+ q0
)/2}2 � (c|p⇥ q|/g)2 and
H1
(p, q) =
✓
1 + cp
0
+ q0
2(H
2
(p, q))�1 + cp
0
+ q0
2(H
2
(p, q))�2
◆
(H2
(p, q))�1 .
Further,
H2
(p, q) =cp
s
2g|p� q| = c|p� q|
s
g2 + 4c2
4g2
.
See Lemma 3.1 [p.316] of [32]. Therefore, H2
(p, q) � c2
|p� q|+ c. This completes our
discussion of the Hilbert-Schmidt form for the linearized collision operator.
186
5.7. Relativistic Vlasov-Maxwell-Boltzmann equation
One might want to prove a stability theorem for the Vlasov-Maxwell-Boltzmann
system similar to our result on the relativistic Landau-Maxwell system [62]. The
relativistic Boltzmann equation (5.1) with any field terms takes the form
@tF + p ·rxF + V (x) ·rpF = C(F, F ).
Now if one wants to apply Guo’s energy method [39] to such a problem, then a key
point is to use high order momentum derivatives. But the momentum derivatives
cause severe problems.
On the one hand, there doesn’t seem to be any analogue in the relativistic case
of the non-relativistic translation invarance of the post-collisional velocities. This
translation invariance is a key point in taking the high order derivatives without
di↵erentiating the post-collisional velocities directly. On the other hand, taking higher
and higher momentum derivatives of the post-collisional velocities in the relativistic
case makes it harder and harder to get estimates for the non-linear term involving
the post-collisional velocities.
More specifically, in the Glassey-Strauss variables (5.14) and (5.16), it was ob-
served by Glassey & Strauss [31] that
|rpq0i|+ |rpp
0i| Cq5
0
�
1 + |p · !|1/21{|p·!|>|p⇥!|3/2}�
.
And furthermore
|rqp0i|+ |rqq
0i| Cq5
0
p0
.
The q growth is not an issue. But still, by the chain rule, |rkpf(p0)| for example is only
bounded above by the derivatives of the function times a large polynomial weight (in
p) which grows larger for larger k. The region where this polynomial growth occurs is
quite small, but not small enough to control an estimate of more than one derivative.
This is the main problem.
In the center-of-momentum variables (5.11) and (5.13), or more generally (5.38),
the problem is essentially the same but it manifests itself in a di↵erent way. Instead of
187
generating polyonmial growth with derivatives, here you generate singularities. And
larger singularities result from taking more derivatives.
This is the key di�culty which prevents me, at the moment, from proving a
stability theorem for the Vlasov-Maxwell-Boltzmann system using Guo’s method.
188
APPENDIX A
Grad’s Reduction of the Linear Boltzmann Collision
Operator
Recall the Boltzmann collision operator
Q(F, G) ⌘Z
R3⇥S2+
B(✓)|u� v|�{F (u0)G(v0)� F (u)G(v)}dud!,
where ✓ is defined by
cos ✓ =u� v
|u� v| · !,
and B(✓) is assumed to satisfy the Grad angular cuto↵ condition
|B(✓)| C |cos ✓| .
Also
S2
+
= {! 2 S2 : ! · (u� v) � 0}.
Further
v0 = v � [(v � u) · !]!, u0 = u� [(u� v) · !]!,(A.1)
|u|2 + |v|2 = |u0|2 + |v0|2,(A.2)
Define the normalized Maxwellian by
µ(v) = (2⇡)�3/2e�|v|2/2.
With the standard perturbation
F = µ + µ1/2f,
we define the linear part of the collision operator by
Lf = �µ�1/2
�
Q(µ, µ1/2f) + Q(µ1/2f, µ)
= ⌫(v)f �Kf.
189
Here K = K2
�K1
and these are defined as [30,34]:
K1
g =
Z
R3⇥S2+
B(✓)|u� v|�µ1/2(u)µ1/2(v)g(u)dud!,
K2
g =
Z
R3⇥S2+
B(✓)|u� v|�µ1/2(u){µ1/2(u0)g(v0) + µ1/2(v0)g(u0)}dud!.
(A.3)
In this Appendix, we will follow [12, 34] to write the integral operator K in a useful
Hilbert-Schmidt form (see also [30,37]).
We will write
Kg =
Z
R3
k(v, ⇠)g(⇠)d⇠.
We notice immediately that
K1
g =
Z
R3
k1
(v, ⇠)g(⇠)d⇠,
where
k1
(v, ⇠) =
Z
S2+
B(✓)|⇠ � v|�µ1/2(⇠)µ1/2(v)d! = c1
|⇠ � v|�µ1/2(⇠)µ1/2(v).
In the rest of this Appendix we will focus on K2
.
We use (A.2) to rewrite K2
in (A.3) as
K2
g = µ1/2(v)
Z
R3⇥S2+
B(✓)|u� v|�µ(u)
⇢
g
µ1/2
(v0) +g
µ1/2
(u0)�
dud!.
Rewrite (A.1) as
v0 = v + [(u� v) · !]!, u0 = v + (u� v)� [(u� v) · !]!.
Change variables u� v ! u to obtain
K2
g = µ1/2(v)
Z
R3⇥S2+
B(✓)|u|�µ(u + v)
⇢
g
µ1/2
(v + uk) +g
µ1/2
(v + u?)
�
dud!,
where now cos ✓ = u · !/|u| and we define
uk = {u · !}!, u? = u� uk = ! ⇥ (u⇥ !).
We have just used the formula
w ⇥ (u⇥ v) = (v · w)u� (u · w)v (8u, v, w 2 R3).
190
Now K2
= Kk2
+ K?2
where
K?2
g = µ1/2(v)
Z
R3⇥S2+
B(✓)|u|�µ(u + v)g
µ1/2
(v + u?)dud!,
The first step is to compare the two parts of K2
g.
To this end we write out the angular coordinates precisely as
! =
0
B
B
B
@
sin ✓ cos�
sin ✓ sin�
cos ✓
1
C
C
C
A
, 0 ✓ ⇡
2, 0 � < 2⇡, d! = sin ✓d✓d�,
where u · ! = |u| cos ✓. Then
K?2
g = µ1/2(v)
Z
R3
du
Z
⇡
2
0
b⇤(✓)d✓Z
2⇡
0
d�|u|�µ(u + v)g
µ1/2
(v + u?),
and we have defined b⇤(✓) = sin ✓B(✓). For fixed !, we complete an orthonormal
basis as
!? =
0
B
B
B
@
� cos ✓ cos�
� cos ✓ sin�
sin ✓
1
C
C
C
A
, ! ⇥ !? =
0
B
B
B
@
sin�
� cos�
0
1
C
C
C
A
.
Now we will use the relation u · ! = |u| cos ✓ to consider u · !? and u · (! ⇥ !?).
Sending ✓ ! ⇡2
� ✓ and �! �± ⇡, we have
sin⇣⇡
2� ✓⌘
= cos ✓, cos⇣⇡
2� ✓⌘
= sin ✓,
sin (�± ⇡) = � sin�, cos (�± ⇡) = � cos�.
Under this transformation ! ! !?, hence in this coordinate system
u · !? = |u| sin ✓
Furthermore, by linear algebra,
u = {u · !}! + {u · !?}!? + {u · (! ⇥ !?)}! ⇥ !?.
By orthogonality,
|u|2 = {u · !}2 + {u · !?}2 + {u · (! ⇥ !?)}2.
191
In other words u · (! ⇥ !?) = 0. We conclude
u = {u · !}! + {u · !?}!?.
In this coordinate system we can alternatively define
!? =! ⇥ (u⇥ !)
|! ⇥ (u⇥ !)| .
And still u · !? =p
|u|2 � (u · !)2 = |u| sin ✓. In either case, we observe that
u? = {u · !?}!?.
Further, by earlier calculations we see that the change of variable ✓ ! ⇡2
� ✓ and
�! �± ⇡ sends !? ! !. Restricting to ✓ ! ⇡2
� ✓ and �! �� ⇡ for conveinence,
we have
K?2
g =
Z
R3
du
Z
⇡
2
0
b⇤⇣⇡
2� ✓⌘
d✓
Z
+⇡
�⇡
d�|u|�µ1/2(v)µ(u + v)g
µ1/2
(v + uk).
Over the region [�⇡, 0] we further change variables � ! � + 2⇡ and use periodicity
to get
K?2
g =
Z
R3
du
Z
⇡
2
0
b⇤⇣⇡
2� ✓⌘
d✓
Z
2⇡
0
d�|u|�µ1/2(v)µ(u + v)g
µ1/2
(v + uk).
But notice that by definition
Kk2
g = µ1/2(v)
Z
R3
du
Z
⇡
2
0
b⇤(✓)d✓Z
2⇡
0
d�|u|�µ(u + v)g
µ1/2
(v + uk).
We can therefore write
K2
g = µ1/2(v)
Z
R3
du
Z
⇡
2
0
b(✓)d✓
Z
2⇡
0
d�|u|�µ(u + v)g
µ1/2
(v + uk),
where
b(✓) = b⇤ (✓) + b⇤⇣⇡
2� ✓⌘
.
Further notice that |b(✓)| C |cos ✓ sin ✓|.
This upper bound motivates |uk| = |u · !| = |u|| cos ✓|. Thus,
K2
g = µ1/2(v)
Z
R3
du
Z
⇡
2
0
b(✓)
| cos ✓|d✓Z
2⇡
0
d�|uk||u|1��
µ(u + v)g
µ1/2
(v + uk).
192
Note that (uk, u?) are invariant under the map ✓ ! ⇡ � ✓ and �! �± ⇡:
sin (⇡ � ✓) = sin ✓, cos (⇡ � ✓) = � cos ✓,
sin (�± ⇡) = � sin�, cos (�± ⇡) = � cos�.
So that
! !
0
B
B
B
@
� sin ✓ cos�
� sin ✓ sin�
� cos ✓
1
C
C
C
A
= �!.
By defining b(✓) = b(⇡ � ✓) for ✓ 2 [⇡/2, ⇡] we can therefore write
K2
g = µ1/2(v)
Z
R3
du
Z ⇡
0
2b(✓)
| cos ✓|d✓Z
2⇡
0
d�|uk||u|1��
µ(u + v)g
µ1/2
(v + uk).
Next we will reduce this expression to Hilbert-Schmidt form.
To this end, expand the exponent of the exponentials as
1
2|v|2 + |u + v|2 � 1
2|v + uk|2 =
1
2|v|2 + |u?|2 + 2u? · (v + uk) +
1
2|v + uk|2
Next define
⇣ = v +1
2uk.
Further split ⇣ = ⇣k + ⇣? where
⇣k = (v · !)! +1
2uk, ⇣? = v � (v · !)!.
And now, since u? · ⇣ = u? · (v + uk) and v + uk = 2⇣ � v, we further expand the
exponent as
1
2|v|2 + |u?|2 + 2u? · ⇣ +
1
2|2⇣ � v|2 = |v|2 + |u?|2 + 2u? · ⇣ + 2|⇣|2 � 2⇣ · v
= |v|2 + |u? + ⇣?|2 + |⇣|2 + |⇣k|2 � 2⇣ · v
= |u? + ⇣?|2 + |⇣k|2 + |v � ⇣|2
= |u? + ⇣?|2 + |⇣k|2 +1
4|uk|2.
Plugging this in, we can write K2
g as
(2⇡)�3/2
Z
R3
du
Z ⇡
0
2b(✓)
| cos ✓|d✓Z
2⇡
0
d�|uk||u|1��
e�12 |u?+⇣?|2� 1
2 |⇣k|2� 18 |uk|2g(v + uk).
193
Equivalently,
K2
g = (2⇡)�3/2
Z
R3⇥S2
dud!2b(✓)
sin ✓| cos ✓||uk||u|1��
e�12 |u?+⇣?|2� 1
2 |⇣k|2� 18 |uk|2g(v + uk).
The last step is to rearrange the integration regions.
Let O = [!1
? !2
? !] be a rotation matrix. Change variables u ! Ou to see that
uk = {u · !}! ! u3
!, u? = u� uk ! u1
!1
? + u2
!2
?.
Define
⇠ = u3
!, ⇠? = u1
!1
? + u2
!2
?.
Split the du = du1
du2
du3
integration into
d⇠? = du2
du3
,
d|⇠| = du1
.
First integrate over d⇠? to obtain
K2
g =
Z
R⇥S2
k⇤2
(v, ⇠)g(v + ⇠)|⇠|d|⇠|d!,
where
k⇤2
(v, ⇠) = (2⇡)�3/2e�18 |⇠|2� 1
2 |⇣k|2Z
R2
2b(✓)
sin ✓| cos ✓| |u|��1e�
12 |⇠?+⇣?|2du?
Since we are integrating over the full sphere, symmetry of the transformation ! ! �!
allows us to write
K2
g = 2
Z 1
0
Z
S2
k⇤2
(v, ⇠)g(v + ⇠)|⇠|d|⇠|d!.
Moreover, the standard transformation from polar coordinates is
|⇠|d|⇠|d! =|⇠|2d|⇠|d!
|⇠| =d⇠
|⇠| .
Here d⇠ now represents integration over R3. We can now write the Hilbert-Schmidt
form for K2
as
K2
g =
Z
R3
k2
(v, ⇠)g(v + ⇠)d⇠
194
where
k2
(v, ⇠) = 4e�
18 |⇠|2� 1
2 |⇣k|2
(2⇡)3/2|⇠|
Z
R2
�
|⇠|2 + |⇠?|2�
��12 e�
12 |⇠?+⇣?|2 b(✓)
sin ✓| cos ✓|d⇠?.
Here given ⇠, the idea is to extend it to an orthonormal basis for R3 denoted by
{⇠1, ⇠2, ⇠/|⇠|}. Then d⇠? = d⇠1
?d⇠1
? and in the formulas above
⇠? = ⇠1
?⇠2 + ⇠3
?⇠2
where above ⇠1, ⇠2 are vectors and ⇠1
?, ⇠2
? are scalars. Moreover,
⇣ = ⇣(v, ⇠) = ⇣k + ⇣?
where
⇣k =(v · ⇠)⇠|⇠|2 +
1
2⇠, ⇣? = v � (v · ⇠)⇠
|⇠|2 = (v · ⇠1)⇠1 + (v · ⇠2)⇠2.
This completes Grad’s reduction.
195
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