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Large BV solutions to the compressible isothermal Euler–Poisson equations with spherical

symmetry

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2006 Nonlinearity 19 1985

(http://iopscience.iop.org/0951-7715/19/8/012)

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INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY

Nonlinearity 19 (2006) 1985–2004 doi:10.1088/0951-7715/19/8/012

Large BV solutions to the compressible isothermalEuler–Poisson equations with spherical symmetry

Dehua Wang1 and Zejun Wang2

1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA2 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic ofChina

E-mail: dwang@math.pitt.edu and zjwang@fudan.edu.cn

Received 10 April 2006, in final form 20 June 2006Published 17 July 2006Online at stacks.iop.org/Non/19/1985

Recommended by A L Bertozzi

AbstractThe Euler–Poisson equations for isothermal flows in charge transport arestudied. The global weak entropy solution with spherical symmetry isconstructed for arbitrarily large initial data in BV. The global solution is inBV and has large total variations. One difficulty is that the system of Euler–Poisson equations is a hyperbolic–elliptic coupled system, and the propertyof finite propagation speed does not hold. A modified Glimm scheme isused to construct the approximate solution of the Euler equations while thePoisson equation is approximated by an integral of the solution to the Eulerequations. Other difficulties include the geometrical structure, the dampingand some additional source terms, which make the analysis more complicatedand difficult. To overcome these difficulties, the solution of a discrete ordinarydifferential equation is used to construct the approximate solution. It is provedthat vacuum will not occur in the solution, which is important to obtain thecompactness of the approximate solution in BV.

Mathematics Subject Classification: 35L65, 76N10, 65M12, 78A35

1. Introduction

The Euler–Poisson equations that arise in modelling charge transport in a self-consistent electricfield have the following form in d-space dimensions:

nt + ∇ · (nu) = 0,

(nu)t + ∇ · (nu ⊗ u + pI) = n∇φ − σnu,

�φ = n − N (x),

(1.1)

where x ∈ Rd is the space variable, t is the time variable, n is the density of the flow of

electrons (or ions), u ∈ Rd is the velocity of the flow, φ is the electric potential, ∇φ is the

0951-7715/06/081985+20$30.00 © 2006 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1985

1986 D Wang and Z Wang

electric field, σ � 0 is related to the relaxation and damping, N is the doping profile or thedensity of a fixed background charge, p = p(n) is the pressure, I denotes the d × d identitymatrix and � denotes the Laplacian in R

d .The Euler–Poisson system (1.1) is classical in plasma physics. This system and its

variants have been used to describe the dynamic behaviour of many important physicalflows, including the propagation of electrons in submicron semiconductor devices [12],the biological transport of ions for channel proteins [2], the motion of a plasma ([7] andreferences therein), the motion of a plasma sheath ( [15] and references therein), the motionof stars in the theory of general relativity [1] and so on. In the hydrodynamic model forsemiconductor devices, n is the electron density. The biological model describes the transportof ions between the extracellular side and the cytoplasmic side of the membrane in theion channels, which are porous proteins inserted across cell membranes, where n is theion concentration. A plasma is a collection of moving electrons and ions. For a plasmaat high frequencies, the electrons and ions tend to move independently, charge separationsoccur, and thus the ions merely provide a uniform background of positive charge. In thiscase, n is the density of electrons. Plasma sheaths arise from the breakdown of the quasi-neutrality in ion–electron plasmas, where n is the density of ions. The Euler–Poisson systemis also used to model the motion of stars consisting of compressible perfect fluids with self-gravitation [1], where n is the density of the fluid and φ is the gravitational potential. Therehave been many studies of the Euler–Poisson equations, see [2,3,7,10,12,14–16] and theirreferences.

In this paper, we are interested in the spherically symmetric problem of the Euler–Poissonequations. We are particularly concerned with the following issues: (a) whether the sphericalsymmetry of the initial state is preserved globally in time; (b) if the initial data are in the BVfunctional space with large total variation, whether the solution is still in the BV space with finitetotal variation at any time. These are difficult questions in general even when the initial datahave small total variation and there is no electric field (i.e. for the Euler equations). We considerthe spherically symmetric solutions of the Euler–Poisson equations (1.1) when the flow isisothermal, i.e. the pressure is a linear function of the density. The one-dimensional problemwas studied in [14] for the Euler–Poisson system. The spherically symmetric problem for theEuler equations in the isothermal case has been studied in [8, 11] (also see their references),and the one-dimensional problem was studied in [13]. The Lagrangian coordinates were usedin [8,11,13]. Besides the geometrical source terms, the difficulty of the spherically symmetricproblem for the Euler–Poisson equations includes the nonlocal effect from the electricfield.

We will construct the global solution of spherical symmetry in Eulerian coordinates usinga modified Glimm scheme [6]. Some ideas in [5, 9, 14, 17] will be adopted to handle theelectric field and other source terms. We first introduce some transformations to reformulatethe problem, through which the continuity equation is changed into the conservation form. Todeal with the momentum equation with source terms, we use a discrete ordinary differentialequation to approximate it, where the damping, the electric field and the other source termare taken into account at the same time. For the Poisson equation, since it is elliptic, we usean approximate integral to construct the approximate solution. It is interesting that, underappropriate conditions, the density and the electric field in the new variables keep the finitepropagation property, while the velocity does not have finite propagation speed. In the originalvariables, all these quantities decay with respect to the radius. Besides, under the conditionsof this paper, we can show that there is no vacuum appearing in the approximate solution. Thisis important since otherwise the L∞ and BV bounds of the approximate solution may not beobtained.

Compressible isothermal Euler–Poisson equations 1987

The paper is organized as follows. In section 2, we reformulate the problem and state themain result. For the reader’s convenience, we also give formal derivations of some propertiesof the solution which lead us to construct the approximate solution. In section 3, we recall somebasic results about the Riemann problems and interaction of the elementary waves. Section 4is devoted to the construction of the approximate solution. In section 5, we will establish theuniform bounds in L∞ and BV, and thus prove the compactness of the approximate solution.In section 6, we will prove that the approximate solution is convergent and the limit is a weakentropy solution.

2. Main results

We consider the global solution of (1.1) with spherical symmetry. Set r = |x|. Assumethat both functions N = N (x) = N (r) and σ = σ(x) = σ(r). The spherically symmetricsolution of (1.1)

n(t, x) = n(t, r), u(t, x) = u(t, r)xr, φ(t, x) = φ(t, r)

satisfies

nt + (nu)r +d − 1

rnu = 0,

(nu)t + (nu2 + p(n))r +d − 1

rnu2 = nφr − σnu,

φrr +d − 1

rφr = n − N (r).

(2.1)

We assume that the compressible flow is isothermal, i.e. p(n) = c2n for some constant c > 0of sound speed. Introduce the new variables

ρ = rd−1n, E = rd−1φr, N = rd−1N .

Then (2.1) becomes

ρt + (ρu)r = 0,

(ρu)t + (ρu2 + p(ρ))r = ρc2 + E

r− σρu,

Er = ρ − N(r),

(2.2)

or, setting m = ρu,

ρt + mr = 0,

mt +

(m2

ρ+ p(ρ)

)r

= ρc2 + E

r− σm,

Er = ρ − N(r),

(2.3)

where we take d = 2 without loss of generality. We consider the initial-boundary valueproblem of system (2.2) for (t, r) ∈ [0, ∞)× [1, ∞), subject to the following initial boundaryconditions:

ρ(0, r) = ρ0(r), u(0, r) = u0(r),

u(t, 1) = 0, E(t, ∞) = E∗(t), (2.4)

with compatibility u0(1) = 0. We assume that there exist some constants L > 1, σ∗ � 0,N∗ > 0 and u∗ , such that

limr→∞ u0(r) = u∗,

σ (r) = σ ∗, N(r) = N∗, ρ0(r) = N∗, for r > L. (2.5)

1988 D Wang and Z Wang

Take an entropy and the entropy flux pair (η, q):

η(ρ, m) = m2

2ρ+ c2ρ ln ρ = ρu2

2+ c2ρ ln ρ, (2.6)

q(ρ, m) = m3

2ρ2+ c2m ln ρ = ρu3

2+ c2ρu ln ρ. (2.7)

The main result of this paper is as follows.

Theorem 2.1. Suppose that (2.5) holds and that, there exists a constant ρ > 0, such that, forany r � 1,

σ(r) � 0, N(r) > 0, ρ0(r) � ρ > 0, (2.8)

(ρ0, u0, σ, N) ∈ BV([1, ∞)), E∗(t) ∈ BV([0, ∞)). (2.9)

Then the initial-boundary value problem (2.2) and (2.4) has a global weak entropy solution(ρ, u, E)(t, r) with

(ρ, u, E)(t, ·) ∈ BV([1, ∞)),

satisfying the following properties.

(1) For any function φ(t, r) ∈ C10([0, ∞) × [1, ∞)) with � ≡ supp φ,∫ ∫

�

(ρφt + ρuφr)(t, r)drdt +∫ ∞

1ρ0(r)φ(0, r)dr = 0, (2.10)

∫ ∫�

(ρuφt + (ρu2 + p)φr + ρ

(c2 + E

r− σu

)φ

)(t, r)drdt

+∫ ∞

1ρ0(r)u0(r)φ(0, r)dr −

∫ ∞

0c2ρ(t, 1)φ(t, 1)dt = 0, (2.11)

and

E(t, r) = E∗(t) +∫ r

∞(ρ(t, r) − N(r))dr. (2.12)

(2) For any φ(t, r) ∈ C10([0, ∞) × [1, ∞)) with � ≡ supp φ, φ > 0,∫ ∫

�

(ηφt + qφr + ρu

(c2 + E

r− σu

)φ

)(t, r)drdt

+∫ ∞

1η(ρ0(r), ρ0(r)u0(r))φ(0, r)dr � 0. (2.13)

Moreover, for any given t > 0, there exists a Lt > L, such that

ρ(t, r) = N∗, E(t, r) = E∗(t), for any r > Lt , (2.14)

limr→∞ u(t, r) = e−σ ∗t u∗. (2.15)

We remark that the conclusion of this theorem shows that ρ and E have the property offinite propagation speed if (2.5) holds. However, even if the condition of u0(r) in (2.5) ischanged into

u0(r) = u∗, for r > L,

we still can only get the conclusion in (2.15) and cannot get the conclusion such as u(t, r) =u∗, for r > Lt . Thus u does not have the property of finite propagation.

Compressible isothermal Euler–Poisson equations 1989

It is easy to rewrite the result of theorem 2.1 for the original Euler–Poisson system (2.1)and we omit the details for the sake of conciseness.

For the convenience of the reader, in the remaining part of this section, we will formallydeduce some properties of the solution to (2.2) and (2.3), which can help us to construct theapproximate solutions and to understand some of their properties.

Integrating (2.2)3 from +∞ to r , we have

E(t, r) = E∗(t) +∫ r

Lt

(ρ(t, r) − N(r))dr. (2.16)

From this formula we can also get the value of E(t, r) at r = 1

E(t, 1) = E∗(t) +∫ 1

Lt

(ρ(t, r) − N(r))dr. (2.17)

From (2.2)2, limr→∞(eσ tu)t = 0. Thus the limiting state of the solution at r = ∞ are

ρ(t, +∞) = N∗, E(t, +∞) = E∗(t), u(t, ∞) = e−σ ∗t u∗. (2.18)

From (2.2)1, we have

d

dt

∫ ∞

1(ρ(t, r) − N(r))dr + N∗u∗e−σ ∗t = 0. (2.19)

Throughout this paper, we denote

α(t, r) = e−σ(r)t , α∗(t) = e−σ ∗t

δ(t, r) = 1 − α(t, r)

σ (r), δ∗(t) = 1 − α∗(t)

σ ∗ .

Moreover, we let δ∗(t) ≡ t when σ ∗ = 0. Now, (2.19) yields∫ ∞

1(ρ(t, r) − N(r))dr =

∫ ∞

1(ρ0(r) − N(r))dr − N∗u∗δ∗(t), (2.20)

∫ ∞

1(ρ(t2, r) − N(r))dr =

∫ ∞

1(ρ(t1, r) − N(r))dr + N∗u∗(δ∗(t1) − δ∗(t2)) (2.21)

for any t1 > t2 > 0. Defining

ρ(t, r) = ρ(t, r) − N(r),

we can easily obtain the following estimates

‖ρ(t, ·)‖L1 �∫ Lt

1(ρ(t, r) + N(r))dr, (2.22)

‖ρ(t, ·)‖L1 �∫ Lt

1(ρ0(r) + N(r))dr + N∗|u∗|δ∗(t). (2.23)

In fact, (2.22) is obvious from the expression of ρ(t, r). We can get (2.23) with the help of(2.20) and (2.22) as follows

‖ρ(t, ·)‖L1 �∫ Lt

1(ρ(t, r) − N(r))dr + 2

∫ Lt

1N(r)dr

=∫ Lt

1(ρ0(r) − N(r))dr − N∗u∗δ∗(t) + 2

∫ Lt

1N(r)dr

=∫ Lt

1(ρ0(r) + N(r))dr − N∗u∗δ∗(t).

1990 D Wang and Z Wang

Moreover, from (2.17) and (2.21), we have

E(t2, 1) − E(t1, 1) = E∗(t2) − E∗(t1) + N∗u∗(δ∗(t2) − δ∗(t1)), (2.24)

and from (2.21), we obtain∫ Lt2

1(1 + ρ(t2, r))dr

=∫ Lt2

1(1 + N(r))dr +

∫ Lt1

1(1 + ρ(t1, r))dr −

∫ Lt1

1(1 + N(r))dr

+N∗u∗(δ∗(t1) − δ∗(t2))

=∫ Lt1

1(1 + ρ(t1, r))dr + (N∗ + 1)(Lt2 − Lt1) + N∗u∗(δ∗(t1) − δ∗(t2)). (2.25)

In the construction of an approximate solution, we will use (2.18) to define the value of anapproximate solution at infinity. To give the definition of ρ (2.25) is used. Then we can use(2.16), (2.17) and (2.24) to define the approximate value of the electric field Eh. We will alsoshow that properties similar to (2.22) and (2.23) hold for the approximate solution, which isimportant to control the total variation of the approximate electric field Eh. This is differentfrom the Euler equations which is strictly hyperbolic.

3. Riemann problems

In this section, we will review some basic facts about the homogeneous Euler equations of (2.2):{ρt + (ρu)r = 0,

(ρu)t + (ρu2 + p)r = 0,(3.1)

where p = c2ρ. Denote V (t, r) = (ρ(t, r), u(t, r)). Consider the classical Riemann initialvalue problem for (3.1) with data

V (0, r) = V0(r) ={

V − = (ρ−, u−), r < r0,

V + = (ρ+, u+), r > r0,(3.2)

and the Riemann initial boundary value problem with data{(ρ(0, r), u(0, r)) = (ρ+, u+), r > 1,

u(t, 1) = 0, t > t0.(3.3)

The eigenvalues of system (3.1) are

λ1 = u − c, λ2 = u + c, (3.4)

and the two Riemann invariants are

w = u + η, z = u − η, (3.5)

where η = c ln ρ. In the w–z plane, the 1-shock polar S1 starting from (w0, z0) can be expressedin the form

w − w0 = f (z − z0), for z � z0. (3.6)

The 2-shock polar S2 can also be expressed in the form

z − z0 = f (w − w0), for w � w0. (3.7)

Here, the function f (x) satisfies

0 � f ′(x) < 1, f ′′(x) � 0, limx→−∞ f (x) = 1. (3.8)

Compressible isothermal Euler–Poisson equations 1991

Therefore, on the w-z plane the two shock polars are symmetric with respect to the diagonal andtheir figures are independent of the starting points. With this special property, Nishida [13] gavethe existence of global BV solutions with arbitrary large initial data. Meanwhile, the invariantregion was proposed in [13]. In particular, the following two lemmas give the solvability andthe invariant regions of the two Riemann problems above.

Lemma 3.1. There exists a unique piecewise smooth entropy solution V (t, r) for the Riemanninitial value problem (3.1) and (3.2) satisfying

w(V (t, r)) = w(ρ(t, r), u(t, r)) � max(w(V −), w(V +)),

z(V (t, r)) = z(ρ(t, r), u(t, r)) � min(0, z(V +)).

For the Riemann initial boundary value problem (3.1) and (3.3), there also exists a globalentropy solution satisfying

w(V (t, r)) = w(ρ(t, r), u(t, r)) � max(w(V +), −z(V +)),

z(V (t, r)) = z(ρ(t, r), u(t, r)) � min(0, z(V +)).

Lemma 3.2. For the Riemann initial value problem, the region

= {(ρ, u) : w � max(w0, −z0), z � z0}is invariant. For the Riemann initial boundary value problem, the region

= {(ρ, u) : w � max(w0, −z0), z � z0}, z0 � 0 � w0 + z0

2,

is invariant. That is, if the Riemann initial data lies in , then the solution of the Riemannproblem and its average also lie in .

The Riemann solution of problem (3.1) and (3.2) is a self-similar solution in variableξ = (r − r0)/t , and it consists of two families of elementary waves (shock wave or rarefactionwave) corresponding to two characteristic fields, respectively. There exists an intermediateconstant state V0, such that V − and V0 can be connected by a 1-wave, and V0 and V + can beconnected by a 2-wave. For problem (3.1) and (3.3), there is only one wave which belongs tothe second family.

With the left state (ρ−, u−) and right state (ρ+, u+), the two shock polars and tworarefaction curves are given by

S1 : u+ − u− + c

(√ρ+

ρ− −√

ρ−

ρ+

)= 0, ρ+ > ρ−, (3.9)

S2 : u+ − u− − c

(√ρ+

ρ− −√

ρ−

ρ+

)= 0, ρ+ < ρ−, (3.10)

R1 : u− + c ln ρ− = u+ + c ln ρ+, ρ+ < ρ−, (3.11)

R2 : u− − c ln ρ− = u+ − c ln ρ+, ρ+ > ρ−. (3.12)

As in [14], we define

F(t) =

t, for t � 0,

2c sinht

2c, for t > 0.

(3.13)

1992 D Wang and Z Wang

Then the two curves R1 and S1 (or R2 and S2) can be represented uniformly as

R1 and S1 : u+ − u− + F(η+ − η−) = 0;R2 and S1 : u+ − u− − F(η+ − η−) = 0.

Therefore, the relation between the intermediate state V0 and the two given states V − and V +

can be expressed by the following formula:

u− − u+ = F(η0 − η−) + F(η0 − η+). (3.14)

Since F(t) in strictly increasing, this equation determines η0 uniquely. When no vacuumappears, we can use the following quantity to measure the strength of the elementary waves

S(V −, V +) = |η− − η0| + |η0 − η+|, (3.15)

where η0 = c ln ρ0 and (ρ0, u0) is the intermediate state in solving Riemann problem (3.1)and (3.2) with initial data (V −, V +). Since η is monotone across a simple wave, we have

S(V −, V +) = T V (|η(·, t)|), t > 0. (3.16)

Next we state more properties of the functional S(V −, V +) since it can be used to measure thetotal variation of the approximate solution. In particular, the following two lemmas [14] areimportant in obtaining local interaction estimates of the elementary waves.

Lemma 3.3. Let V1, V2, V3 be three arbitrary states in {R+\{0}} × R. Then we have

S(V1, V3) � S(V1, V2) + S(V2, V3). (3.17)

In particular, when V2 is the intermediate state of the solution V (ξ) to the Riemann problemwith initial data V1 and V3, i.e.

V2 = V (ξ0), for some ξ0 ∈ R, (3.18)

then

S(V1, V3) = S(V1, V2) + S(V2, V3). (3.19)

Lemma 3.4. Let V ± = (ρ±, u±) be two constant states and

V ± = (ρ±, u± + �±), �± ∈ R,

V ± = (ρ±, αu±), α ∈ (0, 1),

then

S(V −, V +) � S(V −, V +) + |�− − �+|, (3.20)

S(V −, V +) � S(V −, V +). (3.21)

In contrast to the regular procedure in [6], the advantage of the above estimates is thatthere will be no higher order terms in the corresponding Glimm functional.

4. Construction of an approximate solution

We now construct the approximate solution by using a modified Glimm scheme. The meshstep for the spatial variable r is taken to be h. In order to take the effect of symmetry andelectric field into account, we will modify the value of approximate solution at each time step.

Denote rj = 1 + jh and

Ak,j = {tk} × (rj−1, rj+1), k = 1, 2, . . . , j = 1, 3, . . . , (4.1)

Compressible isothermal Euler–Poisson equations 1993

where the time sequence {tk} is defined by

tk =∑l�k

τl, βkτk = h, βk > 0, k ∈ N. (4.2)

Here {βk}∞k=0 is a positive increasing sequence and will be given later.The given functions N(r) and σ(r) are approximated by the following piecewise constant

functions

Nh(r) = N(rj ), σ h(r) = σ(rj ), rj−1 � r < rj+1, j is odd. (4.3)

Choose an equi-distributed sequence {ak}∞k=1, ak ∈ (0, 1). For j is odd, denote

ck,j = akrj−1 + (1 − ak)rj+1

for k � 1 and c0,j = rj . Set U = (ρ, u, E) and V = (ρ, u) and use Uh = (ρh, uh, Eh) andV h = (ρh, uh) to denote the approximate solution.

The approximate values of Eh(tk, r) at r = ∞ and r = 1 are defined as follows:

Eh(tk, ∞) = E∗(tk), (4.4)

Eh(tk, 1) = Eh(tk−1, 1) + E∗(tk) − E∗(tk−1) + N∗u∗(δ∗(tk−1) − δ∗(tk)). (4.5)

The definition of Eh(0, 1) is the same as in (4.7). Here (4.5) is the discretized version of (2.24).Before continuing the construction, we have the following estimates immediately.

Lemma 4.1. Assume that E∗(t) ∈ L∞loc[0, ∞), then for any given T > 0, there exists a positive

constant ET , such that for any k with tk < T , we have

|Eh(tk, ∞)| < ET , |Eh(tk, 1)| < ET .

Proof. The first inequality is obvious. From (4.5), we have

|Eh(tk, 1) − E∗(tk)| � |Eh(tk−1, 1) − E∗(tk−1)| + N∗|u∗||δ∗(tk) − δ∗(tk−1)|,� |Eh(tk−1, 1) − E∗(tk−1)| + N∗|u∗|τk.

By induction on k, we obtain

|Eh(tk, 1) − E∗(tk)| � |Eh(0, 1) − E∗(0)| + N∗|u∗|tk,which implies the second inequality. �

Next we define the approximate solution at t = 0 as

V h(0, r) = V (0, rj ), (t, r) ∈ A0,j , (4.6)

Eh(0, r) = E∗(0) +∫ rj

L

(ρh(0, r) − Nh(r))dr, (t, r) ∈ A0,j . (4.7)

For any k ∈ N , suppose that the approximate solution has been defined for t � tk , and assumethat the following properties hold

ρh(tk, r) > 0, for r > 1, (4.8)

ρh(tk, r) = N∗, E(tk, r) = E∗, for r > L + kh, (4.9)

V h(tk, r) = V h(tk, rj ), in Ak,j , (4.10)

limr→∞ uh(tk, r) = uh(tk, ∞) = e−σ ∗tk u∗. (4.11)

1994 D Wang and Z Wang

We now define Uh(t, r) in the interval [tk, tk+1). Since Uh(tk, r) is piecewise constant and maybe discontinuous at the points (tk, rj+1), j = 1, 3, . . . ., we solve the Riemann initial valueproblem of system (3.1) with initial data

V (tk, r) ={

V − = V h(tk, rj ), r < rj+1,

V + = V h(tk, rj+2), r > rj+1,(4.12)

while near the boundary r = 1, we solve the Riemann initial boundary value problem of system(3.1) with the following conditions

V (tk, r) = V h(tk, r1), r > 1,

u(t, 1) = 0, t � tk. (4.13)

As reviewed in the previous section, the solution to the problem (3.1) and (4.12) consists oftwo simple waves and the solution to (3.1) and (4.13) includes only one simple wave whichbelongs to the second family. We denote the solution to each of the two Riemann problemsby V h

0 (t, r) = (ρh0 (t, r), uh

0(t, r)), tk � t < tk+1. We use a piecewise constant function toapproximate V h

0 (t, r), as usual, in the Glimm scheme:

ρh(tk+1, r) = ρh0 (tk+1−, ck+1,j ), (t, r) ∈ Ak+1,j , j � 1,

uh(tk+1, r) = uh0(tk+1−, ck+1,j ), (t, r) ∈ Ak+1,j , j � 1. (4.14)

For simplicity, we denote all the constant states determined by (4.14) at t = tk as V 0k , V 1

k , . . ..In particular, V 0

k is the state near r = 1, which can be connected to V 1k by a 2-wave by solving

problem (3.1) and (4.13), namely,

Vj

k = (ρ(tk, rj−1), u(tk, rj−1)), for j � 1, j is odd, u0k = 0,

and ρ0k is given by the following equation

u1k = F(c ln ρ1

k − c ln ρ0k ).

Next we define the approximate solution V h(t, r) for tk < t � tk+1 by modifying thevalues on t = tk+1. First, ρh(tk+1, r) is not changed, i.e. define

ρh(t, r) = ρh(t, r), tk < t � tk+1. (4.15)

We use (2.25) to define the approximate value of ρh(tk+1, r). Discretize (2.25) as∫ L+(k+1)h

1(1 + ρh(tk+1, r))dr

=∫ L+kh

1(1 + ρh(tk, r))dr + (N∗ + 1)h + N∗u∗(δ∗(tk) − δ∗(tk+1)). (4.16)

Denote the term on the left-hand side as δk+1, then (4.16) can be simply written as

δk+1 = δk + N∗u∗(δ∗(tk) − δ∗(tk+1)) + (N∗ + 1)h. (4.17)

Thus we define

ρh(tk+1, r) ={

0, r > L + (k + 1)h,

(1 + ρh(tk+1, r))γk+1 − 1 − Nh(r), 1 < r � L + (k + 1)h.(4.18)

Here the correction term γk+1 is given by

δ0 =∫ L

1(1 + ρh(0, r))dr, (4.19)

δk+1 = δk + N∗u∗(δ∗(tk) − δ∗(tk+1)) + (N∗ + 1)h, (4.20)

γk+1 = δk+1 ×(∫ L+(k+1)h

1(1 + ρh(tk+1, r))dr

)−1

. (4.21)

We note that (4.20) is similar to (4.17), and we have the following estimates immediately.

Compressible isothermal Euler–Poisson equations 1995

Lemma 4.2. Assume that βk > |u∗|, then for any given T > 0, if tk � T ,

(1) 0 < δk � δ0 + N∗|u∗|T + (1 + N∗)kh, (4.22)

(2)

∫ ∞

1ρh(tk+1, r)dr =

∫ ∞

1ρh(tk, r)dr − N∗u∗(δ∗(tk+1) − δ∗(tk)), (4.23)

(3)

∫ ∞

1|ρh(tk, r)|dr � ‖ρ0‖L1 + N∗|u∗|T + 2(1 + ‖Nh‖∞)(L + kh − 1). (4.24)

Proof.

(1) Since δ0 > 0, h = βk+1τk+1 and 0 < δ∗(tk+1) − δ∗(tk) < τk+1, we have

δk+1 − δk = N∗u∗(δ∗(tk) − δ∗(tk+1)) + N∗h + h � N∗τk+1(βk+1 − |u∗|),then we know that δk+1 > δk when βk+1 > |u∗|, and thus δk > 0. When tk � T , from thedefinition of δk , we have δk = δ0 − N∗u∗δ∗(tk) + (N∗ + 1)kh. Thus (4.22) follows sinceδ∗(tk) � tk .

(2) From (4.18) and (4.21), we know that∫ ∞

1ρh(tk+1, r)dr = δk+1 −

∫ L+kh

1(1 + Nh(r))dr.

Therefore∫ ∞

1ρh(tk+1, r)dr =

∫ ∞

1ρh(tk, r)dr + δk+1 − δk −

∫ L+(k+1)h

L+kh

(1 + Nh(r))dr

=∫ ∞

1ρh(tk, r)dr + N∗u∗(δ∗(tk) − δ∗(tk+1)).

(3) From the definition (4.18) of ρh(tk, r), we can easily get

|ρh(tk, r)| � ρh(tk, r) + 2Nh(r),

then ∫ ∞

1|ρh(tk, r)|dr =

∫ ∞

1ρh(tk, r)dr + 2

∫ L+kh

1Nh(r)dr.

From (4.23), we get∫ ∞

1ρh(tk, r)dr � ‖ρ0‖L1 + N∗|u∗|δ∗(tk),

and thus inequality (3) follows.

�

Motivated by the discretized version of (2.16), the approximate electric field is defined by

Eh(tk+1, r) = Eh(tk + 1, rj ) = E∗(tk+1) +∫ rj

∞ρh(tk+1, r)dr, (t, r) ∈ Ak+1,j . (4.25)

From this definition, we know that

Eh(tk+1, r) = E∗(tk+1), r > L + (k + 1)h. (4.26)

Finally, we define uh(tk+1, r) by

uh(tk+1, r) = uh(tk+1, r)α(τk+1, rj ) +(c2 + Eh(tk+1, r))

rj

δ(τk+1, rj ), r ∈ Ak,j . (4.27)

1996 D Wang and Z Wang

We can verify that

limr→∞ uh(tk+1, r) = e−σ ∗tk+1u∗.

From (4.15) and (4.26), we know that ρh and Eh have the property of finite propagation speed.However, we do not have the same property for uh due to the influence of the electric field andthe damping. In the special case when E∗(t) ≡ −c2, σ (r) ≡ 0, it is easy to verify that uh hasa finite propagation speed.

For convenience, we denote all the constant states in V h(t, r) at t = tk by V 0k , V 1

k , . . .,that is,

Vj

k = (ρ(tk, rj−1), u(tk, rj−1)), for j � 1, j is odd

and u0k = 0, while ρ0

k is given by the following equality:

u1k = F(c ln ρ1

k − c ln ρ0k ).

This completes the construction of the approximate solution.

5. Uniform estimates

For simplicity of notations, we denote by Vk = (ρk, uk) the approximate solution in the strip[tk, tk+1) constructed in section 4. Let V

j

k , j = 0, 1, . . . be the piecewise constant approximatesolution on the line t = tk defined in section 4. Define

J (Vk) =∞∑

j=0

S(Vj

k , Vj+1k ). (5.1)

It measures the total variation of function ln ρ on the line t = tk .

Lemma 5.1. For the sequence of functions defined above, we have

J (Vk+1) � J (Vk). (5.2)

Proof. We only consider the case when ak+1 � 12 since the other case can be considered

similarly. From lemma 3.3, we have

S(Vj

k , Vj+1k ) = S(V

j

k , Vj+1k+1 ) + S(V

j+1k+1 , V

j+1k ), j � 1,

S(V 0k , V 1

k ) = S(V 0k , V 1

k+1) + S(V 1k+1, V

1k ).

Adding the identities for j ∈ Z, we have

J (Vk) =∞∑

j=0

S(Vj

k , Vj+1k )

= S(V 0k , V 1

k+1) +∞∑

j=1

(S(Vj

k+1, Vj

k ) + S(Vj

k , Vj+1k+1 ))

� S(V 0k+1, V

1k+1) +

∞∑j=1

S(Vj

k+1, Vj+1k+1 ),

= J (Vk+1).

Here we have used the relation V 0k = V 0

k+1. The proof is complete. �

Compressible isothermal Euler–Poisson equations 1997

Lemma 5.2. For the increasing sequence {βk}∞k=1 and any given T > 0, if tk < T , the followingestimates hold.

(1) T V (Ek+1) � ‖ρ0‖L1 + N∗|u∗|T + 2(1 + ‖Nh‖∞)(L + kh − 1),

(2) ‖Ek+1‖∞ � ‖ρ0‖L1 + N∗|u∗|T + 2(1 + ‖Nh‖∞)(L + kh − 1) + ET ,

(3) ‖ηk‖∞ � | ln N∗| + J (Vk),

(4) ‖uk‖∞ � A + BT + J (Vk) + 2(1 + ‖Nh‖∞)khtk,

(5) ‖uk‖∞ � A + BT + J (Vk) + 2(1 + ‖Nh‖∞)khtk,

where

A = ‖u0‖∞ + ‖η0‖∞ + ln |N∗|,BT = [2c2 + ‖ρ0‖L1 + N∗|u∗|T + 2(1 + ‖Nh‖∞)(L − 1) + ET ]T .

Proof. Using (4.24), by direct computation we get

T V (Ek+1) =∑j :odd

|Eh(tk+1, rj ) − Eh(tk+1, rj+2)|

�∫ ∞

1|ρ(tk+1, r)|dr � ‖ρ0‖L1 + N∗|u∗|tk + 2(1 + ‖Nh‖∞)(L + kh − 1),

and (1) follows. Estimate (2) can be easily obtained by using

‖Ek‖∞ � ET + T V (Eh(tk+1, r)).

Estimate (3) is obvious since ‖ηk‖∞ � |ηk|t=∞ + J (Vk).Next we continue to prove (4) and (5). Denote the right-hand side of (2) in this lemma

as E∞. We only discuss the case that r � r1 since the case that near r = 1 can be discussedsimilarly. Let us first prove the following estimate.

|uk| + ηk � ‖u0‖∞ + ‖η0‖∞ + (c2 + E∞)tk. (5.3)

It holds for k = 0. Suppose that it holds for k, then by the property of invariant region inlemma 3.2, we obtain

|uk+1| + ηk+1 � maxt=tk

(|uk| + ηk) � ‖u0‖∞ + ‖η0‖∞ + (c2 + E∞)tk. (5.4)

By using (4.27), we have

|uk+1| � |uk+1| + (c2 + ‖Ek‖∞)δ(τk+1, r)

� ‖u0‖∞ + ‖η0‖∞ + (c2 + ‖Ek‖∞)tk+1 − ηk+1.

From this inequality we get (5.3) since ηk+1 = ηk+1 and δ(τk+1, r) � τk+1. Then, estimate (4)can be obtained by combining (5.4) and (2), and estimate (5) can be obtained from (5.3) and(2). The proof is complete. �

We note that estimates (1) and (2) in lemma 5.2 give the compactness of Eh which iscontrolled by ‖ρ0‖L1 since it is nonlocal. Next we will prove that J (Vk) is bounded for tk < T .Thus from (3) in lemma 5.2, we know that there is no vacuum appearing in the approximatesolution. Moreover, since kh < βkT , then kh on the right side of the inequalities in this lemmacan be replaced by βkT .

We now prove that J (Vk) is bounded.

1998 D Wang and Z Wang

Lemma 5.3. The functional J (Vk) defined in (5.1) has the following property:

J (Vk+1) � J (Vk) + τk+1T V (σh)J (Vk+1) + (DT βk+1 + FT )τk+1, (5.5)

where

DT = 2T (1 + ‖Nh‖∞)(3 + T V (σh)T ),

FT = 3(c2 + ET + ‖ρ0‖L1 + N∗|u∗|T + 2(1 + ‖Nh‖∞)(L − 1) + (A + BT )T V (σh)).

Proof. Denote

�j

k ≡

c2 + Eh(tk, rj )

rj

δ(τk, rj ), (tk, r) ∈ Ak,j ,

0, r = 1.

Obviously, �(tk, r) ≡ �j

k, (tk, r) ∈ Ak,j and |�(tk, r)| � (c2 + ‖Ek‖∞)τk . For simplicity,denote α

j

k = α(τk, rj ). Then by the construction of the approximate solution, we have

uh(tk+1, rj ) = uh(tk+1, rj )α(τk+1, rj ) + �j

k+1.

Using lemma 3.4, we have

S(Vj

k , Vj+1k ) = S((ρ

j

k , uj

kαj

k + �j

k), (ρj+1k , u

j+1k α

j+1k + �

j

k+1))

� S((ρj

k , uj

kαj

k )), (ρj+1k , u

j+1k α

j+1k )) + |�j

k − �j+1k |

� S((ρj

k , uj

kαj

k ), (ρj+1k , u

j+1k α

j

k )) + S((ρj+1k , u

j+1k α

j

k ), (ρj+1k , u

j+1k α

j+1k )) + |�j

k − �j+1k |

� S(Vj

k , Vj+1k ) + |�j

k − �j+1k | + ‖uk‖∞|αj

k − αj+1k |.

Thus,

J (Vk) � J (Vk) + T V (�(tk, r)) + ‖uk‖∞T V (α(tk, r)). (5.6)

Combining this estimate with lemma 5.1, we get for k ∈ Z

J(Vk+1) � J (Vk) + T V (�(tk+1, r)) + T V (α(τk+1, r))‖uk+1‖∞. (5.7)

But since

|α(τk, r)| � 1, T V (α(τk, r)) � τkT V (σh),

|δ(τk, r)| � τk, T V (δ(τk, r)) � τ 2k T V (σh),

and ‖Ek+1‖∞ � T V (Eh(tk+1, r)), we have

T V (�(tk+1, r))

�∥∥∥∥c2 + Eh(tk+1, r)

r

∥∥∥∥∞

T V (δ(τk+1, r)) + T V(c2 + Eh(tk+1, r)

r

)‖δ(τk+1, r)‖∞

� (c2 + ‖Ek+1‖∞)T V (σh)τ 2k+1 +

(c2 + T V (Eh(tk+1, r)) + ‖Eh(tk+1, r))‖∞

)τk+1

�((c2 + ‖Ek+1‖∞ + ET )(1 + τk+1T V (σh)) + T V (Eh(tk+1, r))

)τk+1

� (c2 + T V (Eh(tk+1, r)) + ET )(2 + τk+1T V (σh))τk+1,

and

T V (α(τk+1, r))‖uk+1‖∞ � ‖uk+1‖∞T V (σh)τk+1.

If we let τk+1T V (σh) < 1, we obtain estimate (5.5) with the help of lemma 5.2. �

Compressible isothermal Euler–Poisson equations 1999

If we assume further that βk is bounded from above, that is

βk � βT , for every tk < T . (5.8)

Then we can deduce the following proposition directly from lemma 5.3.

Lemma 5.4. For any given T > 0, if tk+1 < T and the increasing sequence {βk}∞k=1 satisfiesβk � βT for some positive number βT , then the following estimate holds:

J (Vk+1) � (J (V0) + (DT βT + FT )T ) exp{2 T V (σh) T }, (5.9)

where DT and FT are given by lemma 5.3.

Proof. Denote Hk = J (Vk) exp{−2tkT V (σh)} and H0 = J (V0). From lemma 5.3 we get

(1 − T V (σh)τk+1)Hk+1 exp{2τk+1T V (σh)} � Hk + (DT βT + FT )τk+1 exp{−2tkτk+1T V (σh)}.Since e2s(1 − s) � 1 when s ∈ (0, 1/2), we have

Hk � H0 + (DT βT + FT )T ,

which implies (5.9). �Since J (Vk) is used to measure the total variation of the approximate solution, from

proposition 5.4 we know that J (Vk) is bounded for any given T > 0 if (5.8) holds.Next we define the sequence {βk}∞k=1 which should satisfy two requirements. The first one

is the Courant–Friedrichs–Lewy condition, that is

βk � ‖uk‖∞ + c. (5.10)

The second requirement is that {βk}∞k=1 must be bounded from above such that the sum of allthe τk must be larger than the given T > 0. To this end, we need (5.8) to be satisfied.

Lemma 5.5. There exists a positive increasing sequence {βk}∞k=1, such that the conditions (5.8)and (5.9) can be satisfied.

Proof. Choose constant j0 > J(V0), and define

β1 = |u∗| + c + j0,

βk+1 = c + A + BT + jk + 2(1 + ‖Nh‖∞)khtk. (5.11)

Then jk is recursively defined as follows:

(1 − τkT V (σh))jk = jk−1 + (DT βk + FT )τk. (5.12)

According to lemma 5.3 and (5) of 5.2, we have jk > J (Vk) and (5.10) is satisfied.From the definition of βk , (5.11) and (5.12), we get

(1 − τkT V (σh))βk+1

= jk−1 + (DT βk + FT )τk + (1 − τkT V (σh))(c + A + BT + 2(1 + ‖Nh‖∞)hktk)

= βk − (c + A + BT )τkT V (σh)) + 2(1 − τkT V (σh))(1 + ‖Nh‖∞)khtk

−2(1 + ‖Nh‖∞)(k − 1)htk−1 + (DT βk + FT )τk.

Since

ktk − (k − 1)tk−1 = tk−1 + kτk, kτk < T , h = τkβk,

we then have

(1 − τkT V (σh))βk+1

� (1 + (DT + 4(1 + ‖Nh‖∞)T )τk)βk + ((c + A + BT )T V (σh) + FT )τk. (5.13)

2000 D Wang and Z Wang

Thus using the same procedure as in proving proposition 5.4, we obtain (5.8) when τk+1 issmall. The proof is complete. �

From all of the above estimates, we now have the following theorem, which shows thatthe sequence of the approximate solution and its total variation are bounded uniformly for anygiven T > 0, and thus we obtain the compactness by Helly’s theorem.

Theorem 5.1. Assume that the sequence {βk}∞k=1 is given as in lemma 5.5, then for any fixedT > 0, there exists some positive constant CT , such that

‖Ek‖∞ + T V (Ek) + ‖uk‖∞ + T V (uk) + ‖ηk‖∞ + T V (ηk) � CT . (5.14)

Proof. It remains to verify the uniform bound of T V (uk). If ak < 12 , then

|uh(tk, rj ) − uh(tk, rj+2)| = |F(η0 − η(tk, rj )) + F(η0 − η(tk, rj+2))|.Since F(t) � |t | cosh(t/2c), we obtain

|uh(tk, rj ) − uh(tk, rj+2)|� S(V h(tk, rj ), V h(tk, rj+2)) cosh(S(V h(tk, rj ), V h(tk, rj+2))/2c)

� S(V h(tk, rj ), Vh(tk, rj+2)) cosh(J (Vk)/2c).

Thus we have

T V (uk) � J (Vk) cosh(J (Vk)/2c). (5.15)

The case when ak > 12 can be proved similarly. The proof is complete. �

6. Convergence and consistency of the approximate solution

In this section, we show the convergence and consistency of the approximate solutionconstructed in section 4 and thus prove theorem 2.1.

Proof. From theorem 5.1, the sequence of approximate solution (ρh, uh) has a convergentsubsequence (still labelled) (ρh, uh), that is

ρh → ρ, ρhuh → ρu, Eh → E, a.e. as h → 0. (6.1)

From (4.27),

|uh(tk, r) − u(tk, r)| � C(σh|u(tk, r)| +

c2 + E

rj

)τk,

thus we have

uh → u, a.e. as h → 0. (6.2)

Next we prove that the limit (ρ, u, E) of the approximate solution is a weak entropysolution of problem (2.2) and (2.4). For any fixed T > 0, take a test function φ(t, r) ∈C1

0([0, T ) × [1, +∞)) with � ≡ supp φ. Denote X = �∞k=1(0, 1).

Compressible isothermal Euler–Poisson equations 2001

Using Green’s formula, we have

I =∫ ∫

�

(ρh(t, r)φt (t, r) + ρh(t, r)uh(t, r)φr(t, r)

)drdt +

∫ ∞

1ρh(0, r)φ(0, r)dr

=∑

k

∫ tk

tk−1

ρh(t, 1)uh(t, 1)φ(t, 1)dt

−∑

k

∑j :odd

∫ rj+1

rj−1

(ρh(tk, r) − ρh(tk, r))φ(tk, r)dr

−∑

k

∑j :odd

∫ rj+1

rj−1

(ρh(tk, r) − ρh0 (tk, r))φ(tk, r)dr

≡ I1 − I2 − I3. (6.3)

From the construction of the approximate solution, we have

uh(t, 1) = 0, ρh(tk, r) = ρh(tk, r).

From the standard procedure in proving the convergence of the Glimm scheme (see [6]), weknow that

‖I3‖L2(X) � CT V (ρh(tk, r))h,

then

‖I‖L2(X) � CT V (ρh(tk, r))h → 0, as h → 0, (6.4)

and thus, we have, subject to a subsequence, as h → 0,∫ ∫�

(ρh(t, r)φt (t, r) + ρh(t, r)uh(t, r)φr(t, r))drdt +∫ ∞

1ρh(0, r)φ(0, r)dr → 0. (6.5)

Similarly to (6.3), we have by direct computation,

II =∫ ∫

�

(ρh(t, r)uh(t, r)φt (t, r) + ρh(t, r)((uh)2(t, r) + c2)φr(t, r))drdt

+∫ ∞

1ρh(0, r)uh(0, r)dr

=∑

k

∫ tk

tk−1

(ρh(t, 1)((uh(t, 1))2 + c2)φ(t, 1))dt

−∑

k

∑j :odd

∫ rj+1

rj−1

(ρh(tk, r)uh(tk, r) − ρh(tk, r)u

h(tk, r))φ(t, r)dr

−∑

k

∑j :odd

∫ rj+1

rj−1

(ρh(tk, r)uh(tk, r) − ρh

0 (tk, r)uh0(tk, r))φ(t, r)dr

≡ II1 − II2 − II3.

Since uh(t, 1) = 0, we have

II1 →∫ ∞

0c2ρ(t, 1)φ(t, 1)dr.

As before, the standard procedure of the Glimm scheme [6] gives

‖II3‖L2(X) � CT V (ρhuh)h.

2002 D Wang and Z Wang

Then, subject to a subsequence, we have II3 → 0, a.e. Using the standard procedure of theGlimm scheme [6], we may assumeφ(t, r) = φ(ti, rj ) inAk,j . The term withφ(t, r)−φ(ti, rj )

is convergent to zero as h → 0. From (4.27), we know that

ρh(tk, r)uh(tk, r) − ρh(tk, r)u

h(tk, r)

= ρh(tk, r)(

− σhuh +c2 + Eh

rj

)1 − e−σhτk

σ h

=∫ τk

0e−σhsρh

(− σhuh +

c2 + Eh

rj

)ds

=∫ tk

tk−1

e−σh(s−tk)ρh(

− σhuh +c2 + Eh

rj

)ds

= e−σh(ζk−tk)

∫ tk

tk−1

ρh(

− σhuh +c2 + Eh

rj

)ds,

where ζk ∈ (tk−1, tk). Thus

II2 =∑

k

∑j :odd

e−σh(ζk−tk)

∫ rj+1

rj−1

∫ tk

tk−1

ρh(

− σhuh +c2 + Eh

rj

)φ(t, r)drdt

→∫ ∫

�

ρ

(−σu +

c2 + E

r

)φ(t, r)drdt, as h → 0. (6.6)

Therefore, subject to a subsequence, as h → 0,

II →∫ ∞

0c2ρ(t, 1)φ(t, 1)dt −

∫ ∫�

ρ(

− σu +c2 + E

r

)φ(t, r)drdt. (6.7)

From (6.1), (6.5) and (6.7), and using Lebesgue’s dominated convergence theorem, we knowthat the limit functions (ρ, u, E) satisfy (2.2)1 and (2.2)2.

For the γk constructed in (4.21), we can easily verify that it is uniformly bounded andequi-continuous, then it has a convergent subsequence. According to (4.16), the limit must beequal to 1, i.e. γk → 1, as h → 0. Thus

ρh → ρ − N = ρ.

Therefore, we obtain (2.16) which implies (2.12).Finally, we verify that the entropy condition (2.13) must be satisfied. Denote (ηh, qh) =

(η(ρh, ρhuh), q(ρh, ρhuh)). Here the entropy pair (η, q) is given by (2.6) and (2.7). FromGreen’s formula, we have

III =∫ ∫

�

(ηh(t, r)φt (t, r) + qh(t, r)φr(t, r)

)drdt +

∫ ∞

1ηh(0, r)φ(0, r)dr

=∑

k

∫ tk

tk−1

∑(s[η] − [q])dt −

∑k

∑j :odd

∫ rj+1

rj−1

(ηh(tk, r) − η(tk, r))φ(t, r)dr

−∑

k

∑j :odd

∫ rj+1

rj−1

(η(tk, r) − η0(tk, r))φ(t, r)dr

≡ III1 − III2 − III3, (6.8)

Compressible isothermal Euler–Poisson equations 2003

where the summation∑

in III1 is taken over all the shock waves in the Riemann solutionsbetween tk−1 � t � tk , and [η] and [q] denote the jumps of η and q across the shock wave fromleft to right, respectively. Since s[η] − [q] > 0, then III1 > 0. In III3, η(tk, r) is the Riemannsolution with initial data η0(tk, r). From the standard procedure of the Glimm scheme [6], wehave, subject to a subsequence, III3 → 0 as h → 0. By direct calculations,

ηh(tk, r) − ηh(tk, r) = ρhuh(uh − uh) − 12ρh(uh − uh)2.

Using the same procedure as in (6.6), we have, as h → 0,∑k

∑j :odd

∫ rj+1

rj−1

ρhuh(uh − uh)φ(t, r)dr →∫ ∫

�

ρu(

− σu +c2 + E

r

)φ(t, r)drdt.

In addition, since (eστ − 1)/σ � τ , using the same procedure as in (6.6), we have, as h → 0,∑k

∑j :odd

∫ rj+1

rj−1

1

2ρh(uh − uh)2dr � 1

2τ1

∑k

∑j :odd

∫ rj+1

rj−1

∫ tk

tk−1

eσh(ζk−tk)ρh

×(

− σhuh +c2 + Eh

rj

)2drdt → 0. (6.9)

From (6.8) and (6.9), we conclude that the entropy condition (2.13) holds.The proof of theorem 2.1 is complete. �

Acknowledgments

DW’s research was supported in part by the National Science Foundation and the Office ofNaval Research.

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