The Relaxation Limits of the Two-Fluid Compressible Euler-Maxwell Equations

WANG Shu and ZHENG Lin

College of Applied Science,Beijing University of Technology,Beijing 100124,China.

1 Introduction

In this paper,we consider the three-dimensional two- fl uid(including electrons and ions)Euler-Maxwell equations in a torus T=(R/Z)3：

whereα=e,i,qi=1,qe=-1;neandnistand for the density of the electrons and ions;ueanduistand for the velocity of the electrons and ions;EandBare respectively the electric field and magnetic field;p=p(nα)is the pressure function which is sufficiently smooth and strictly increasing fornα＞0.These variables are functions of a three-dimensional position vectorx∈T and of the timet＞0.In the above systems the physical parameters are the electron massmeand the ion massmi,the momentum relaxation timesτeandτi,and the permittivityεand the permeabilityμ.

For simplicity,we denotemα=1,ε,μ=1 andτe=τi=τ,then we obtain the following systems：

Furthermore,we make the time scaling by replacingtbyand define the enthalpy functionh(nα)by

So the system we considered is rewritten the following reduced two- fluid Euler-Maxwell systems：

with initial data：

The study of compressible Euler-Maxwell equations began in 2000,Chen,Jerome and Wang[1]prove the existence of global weak solutions of the simplified Euler-Maxwell equations by using the method of step by step Godunov scheme combined with compensated compactness;in 2007 and 2008,Peng and Wang[2,3]study the non relativistic limit convergence problem for compressible Euler-Maxwell equations to compressible Euler-Poisson equations and the composite limits of the quasi neutral limit and the non relativistic limit for compressible Euler-Maxwell equations;Peng,Wang and Gu[4]discuss the relaxation limit of compressible Euler-Maxwell equations and the existence of global smooth solution in 2011;in the same year,Wang,Yang and Zhao[5]research the relaxation limit of the plasma two- fluid Euler-Maxwell equations with the help of Maxwell iteration and energy method for the well-prepared initial data;Mohamed-Lasmer Hajjej and Peng[6]study the relaxation limits of the one- fluid Euler-Maxwell equation with initial layer by the method of asymptotic expansion.

For later use in this paper,we recall some inequalities in Sobolev spaces[7,8]and the local existence of smoothsolutions for symmetrizablle hyperbolic equation.For anys＞0,we denote by ‖·‖sthe norm of the usual Sobolev spaceHs(T),and by ‖·‖ and ‖·‖∞the norms ofL2(T)andL∞(T),respectively.

Lemma 1.1.(See[9,10])Let s≥3be an integer and∈Hs(T)withfor some given constant k＞0,independent of τ.Then there exist＞0and a unique smoothsolutionto the periodic problem(1.6)-(1.9)defined in the time interval[0,with

The main result is as follows：

Theorem 1.1.For any integer s≥3,under the Proposition1.1and the following conditions:

here(nα,τ,uα,τ,Eτ,Bτ)(t,x)is the approximate solutions of(1.11)-(1.14),such that as τ→0wehave≥T1and the solutionsatis fi es:

for any t∈[0,Tτ],here Tτ≤T1.

Our main purposein this paperis to studythe relaxation limits of the two- fl uid Euler-Maxwell systems(1.11)-(1.14)with initial data(1.15).We consider the problem with initial layer.In order to establish our result,we make an asymptotic expansion including initial layer.In Section 2,we will take the approximate expression into systems(1.11)-(1.14)such that we get the error estimates of the remainders.In Section 3,we prove the main result about the convergence.

2 Asymptotic expansion

In this section,we make an asymptotic expansion with initial layer functions and take it into the systems,then we can get the expression of the first-order initial layer functions and second-order initial layer functions.Furthermore,we obtain the estimates of the remainders which produced by approximate solutions and extra solutions.

We know that

where(nα,τ,uα,τ,Eτ,Bτ)is the approximate solution,the inner function,and(nα,I,uα,I,EI,BI)is the initial layer function.

Firstly,we make the following ansatz for inner function

and take it into systems(1.11)-(1.14),we obtain

there ?×E0=0 implies the existence of a potentialφ0such thatE0=-?φ0.Thensolves a classical system drift-diffusion equations：

with initial condition：

Then we can get the first order companbility condition：

whereφ0is determined by-△φ0=ni,0-ne,0.It is similar to[6].

Secondly,let the initial data of an approximate solution(nα,τ,uα,τ,Eτ,Bτ)(t,x)have an asymptotic expansion of the form：

where(nα,0,τuα,0,E0,τB0)are the given smoothfunctions solutions;moreover the asymptotic expansion including initial layer correction is

wherez=t/τ2is the fast variable,the subscriptIstands for the initial layer variable andis the correction term defined by：

wherem=0,1,2,....

Obviously,(n0,u0,E0,B0)satisfies the systems(2.2)-(2.5).Putting expression(2.7)into(1.11)and(1.13),we obtainwhich imply

Putting expression(2.7)into(1.14)and using(2.5)and(2.9),we have

which imply

Putting expression(2.7)into(1.12)and using(2.2),we get

From(2.6)-(2.7),we have

together with(2.12),we get the solution about first order initial layers for variableu

Using the similar way,we can obtain the second order initial layers

Suppose that(nα,1,E1,B1)is smooth function,and let

Together with(2.13),(2.14)and(2.16),we have

Similarly,together with(2.14),(2.17)and(2.19),we obtain

There we take

Then we have

From a series of calculations,we get

Then we have

According to the asymptotic expansions above,set

Then we have

Moreover,equations(2.4),(2.8)and(2.24)imply that

and equations(2.5)and(2.27)imply that

define the remaindersand

Because that there isη∈[0,t]?[0,T1],such that

then we have

After a simple calculation,we get

Lemma 2.1.Let s≥3be an integer.For given smooth data,the remaindersandsatify

where C＞0is a constant independent of τ.

3 Proof of the convergence result

In this section we prove the main convergence result from approximate periodic solution to exact solution to two- fluid Euler-Maxwell equations.Letbe the exact solution to(1.11)-(1.14)with initial dataan d(nα,τ,uα,τ,Eτ,Bτ)be an approximate periodic solution defined on[0,T1],with

By Lemma 1.1,the exact solutionis defined in a time intervalwithSinceand the embedding fromHs(T)toC(T)is continuous,we haveFrom(1.17)and(1.18)and assumptionwe deduce that there existand a constantC0＞0,independent ofτ,such that

Similarly,the functiont7→‖((t,·),(t,·),Eτ(t,·),Bτ(t,·))‖sis continuous inC([0,]).From(1.17),the sequence(‖ ((0,·),(0,·),Eτ(0,·),Bτ(0,·))‖s)τ＞0is bounded.Then there exist∈(0,]and a constant,still denoted byC0,such that

Then we defineTτ=min(T1,)＞0,such that the exact solution and the approximate are both defined on[0,Tτ],and the exact solution satisfy：

whereC＞0 is a constant independent ofτ.Obviously it is valid from the paper[6].

Let

obviously the error functionsatis fi es：

Set

whereα=e,i,(e1,e2,e3)is the canonical basis ofR3,yjdenotes theith component ofy∈R3andI3is the 3×3 unit matrix.Then systems(3.3)and(3.4)for unknowncan be rewritten as

It is symmetrizable hyperbolic with symmetrizer

which is a positive definite matrix whenwhereα=e,i.Moreover

is symmetric for all 1≤j≤3,where eachDjis a constant matrix

In order to prove Theorem 1.1,it suffices to establish uniform estimates ofWτwith respect toτ.In what follows,we denote byC＞ 0 various constants independent ofτand foretc.The main estimates are contained in the following two lemmas forrespectively.We fi rst consider the estimate for

Lemma 3.1.Under the assumptions of Theorem1.1,for all t∈(0,Tτ],as τ→0we have

Proof.Differentiating equation(3.7)with respect toxyields and then multiplying byand taking the inner product of the resulting equation withwe obtain

where

From estimating each term of equation(3.10),for?|β|≤s,we have

In tegrating this equationover(0,t)witht∈(0,Tτ)and summing up over all|β|≤s,takingε≥0 suf fi ciently small that the term includingcan be controlled by the left-hand side,together with condition(1.18)for the initial data,we get(3.9).

The estimate foris similar to the paper[6],so we give the simple steps.Now we establish the estimate for,it will be complicated because there exist both electron and ion.

Lemma 3.2.Under the assumptions of Theorem1.1,for all t∈(0,Tτ],as τ→0we have

Proof.For a multi-indexβ∈N3with|β|≤s,differentiating the equations(3.5)and(3.6)with respect tox,we have

and

Taking the inner product of equation(3.15)withFτβand taking the inner product of equation(3.16)withGτβ,then we have

there because

Then we have

Using energy estimate,(1.21)and Lemma 2.1,we obtain

Integrating(3.20)over(0,t),summing up overβsatisfying|β|≤sand using(1.18),we obtain the lemma.

Proof of Theorem 1.1.Letτ→ 0 andε＞ 0 be suf fi ciently small.By Lemma 3.1 and Lemma 3.2,fort∈(0,Tτ]we have

Then it follows from(3.21)that

with

From Gron wall inequality,we get

Therefore,from(3.23)we obtain

By a standard argument on the time extension of smooth solutions,we obtainTτ3≥T1,i.e.Tτ=T1.This fi nishes the proof of Theorem 1.1.

Acknowledgement

The authors thank the referee for various comments which allow to improve the presentation of the paper.

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