(萇生闖) (段然)

School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical,Central China Normal University, Wuhan 430079, China

E-mail: csc981020@163.com; duanran@mail.ccnu.edu.cn

Abstract In this paper,we study the zero dissipation limit with a vacuum for the reacting mixture Navier-Stokes equations.For proper smooth initial data that the initial density tends to zero as the relevant physical coefficients tend to zero,we demonstrate that the solution tends to a rarefaction wave connected to a vacuum on the left side coupled with a zero mass fraction of reactant.What is more,the uniform convergence rate is obtained.

Key words zero dissipation limit;vacuum;reacting mixture;Navier-Stokes equations

1 Introduction and Main Results

This paper is concerned with the zero dissipation limit for the reacting mixture equations[2,34]of compressible,non-isentropic flow

whereρ ≥0,u,θ>0,Zare the density,the fluid velocity,the absolute temperature and the mass fraction of reactant,respectively.The total species energy iswhereestands for the internal energy.The mass fraction of reactantZis uniformly bounded an with upper bound one and a lower bound zero,i.e.,0≤Z ≤1 (the proof can be found in [27]).In addition,the positive constantsεμ,εκ,εd,εqandεKare the coefficients of bulk viscosity,the heat conduction,the coefficients of species diffusion,the difference in the rates of formation of the reactant and the product and rate of the reactant.Also,Ψ(θ) denotes a rate function determined by the Arrhenius law

which is generally a uniformly bounded and discontinuous function.Hereθiis the ignition temperature and A is the activation energy.By the definition,we assume that the functionΨ(θ) satisfies

whereMis a generic positive constant.

We also assume that the gas is ideal,

wherea=Randγ>1 denote the special gas constant and the adiabatic constant,respectively.

By the Gibbs relation

we may deduce from (1.1) the entropy equation

and the temperature equation

The systematic development of the theory of chemical reactions in fluid was first introduced by Williams [34]in 1963.In particular,the conservation equations of combustion were stated there.Ever since then,lots of mathematicians have studied the solutions,structure,and behaviors of combustion through numerical calculations or theoretical study.Gardner [6]and Wagner [33]studied planer and steady combustion waves through a numerical calculations approach.As for theoretical study through partial differential equations,Chen and co-workers made a significant contribution,establishing global-in-time existence theorems for generalized solutions to the compressible Navier-Stokes equations for an initial boundary value problem in[2].They proved the global existence of solutions to the Navier-Stokes equations with large discontinuous initial data in [3].They also obtained the global entropy solutions in [4].In[30],Wang established the existence,uniqueness,and regularity of global solutions with general large initial data.Li [24]gave the proof of the weak existence theorem to the Cauchy problem.The large-time behaviors of the solution were studied in [5,38].In [35],Williams introduced limiting values of various parameters which describe various phenomena in the mixture reaction process.Zhang [40]studied the vanishing species diffusion limit,the rate of reactant limit,and the convergence rates.Here,we study the zero dissipation limit of the model.

In general,the limit inviscid system may contain a singularity such as a shock or a vacuum.Therefore,it is necessary and interesting to justify these phenomena,especially the rarefaction wave with a vacuum.For more on the study of zero disssipation regarding basic waves with no vaccum,we refer readers to [1,9–15,17–20,26,28,31,32,36,37,39,41].For when a vaccum appears,it was pointed out by Liu-Smoller [25]that only rarefaction wave can be connected to the vacuum.Jiu,Wang and Xin [21]considered the large time behavior towards rarefaction waves for isentropic gas.Huang,Li and Wang[16]first proved the inviscid limit for solutions to a rarfaction wave connected to a vacuum for isentropic compressible Navier-Stokes equations.Later,Li and Wang [22]and Wang,Li and Wang [23]generated the result for full compressible Navier-Stokes equations.For other models,Gong studied the zero dissipation limit of the micropolar equations for rarefaction waves ([8]and [7]).

As we consider the initial data for the mass fractionzto be zero,the mass fraction equation(1.1)4indicates thatzequals zero when the dissipation vanishes.Then the compressible Navier-Stokes equations in (1.1) tend to the corresponding Euler equations

A rarefaction wave connected to a vacuum can be constructed by defining the Riemann initial data to (1.3) (see [29]) as

We calculate the eignvalues of system (1.3) as

wherecis a local speed defined by

We also calculate the Riemann invariants,called the 1-Reimann invariants and the 3-Riemann invariant,as

In this paper,we consider the case where the Riemann problem (1.3),(1.4) admits a unique global weak (3-rarefaction wave) solution,and therefore we assume that the initial data of (1.4) lies on the 3-rarefaction wave curve.The initial veocityu-may be defined asdue to the fact that the 3-Riemann invariantis constant in(t,x)along the 3-rarefaction wave curve.Then the 3-rarefaction wave(ρr3,ur3,θr3),(ξ=)connecting the vacuumρ=0 to(ρ+,u+,θ+)is the self-similar solution of (1.3)defined by

The momentummr3=mr3(ξ)and the total internal energyer3=er3(ξ)of a 3-rarefaction wave are defined as

In this paper,we devote to obtain the zero dissipation limit for a rarefaction wave with a vacuum for 1D reacting mixture equations.

Now we prepare to state our main theorem.First,we know thatμ,κ,d,qandKare constants,so the cofficientsεμ,εκ,εd,εqandεKshould be on the same order asε[20,22],so we can take that

Inspired by [16,22,23],by setting the proper initial data (3.1),we construct a sequence of solutions (ρε,mε,:=ρεθε,Zε)(ξ) to the reacting mixture equations (1.1),which converge to the 3-rarefaction wavedefined above asεtends to zero.

Our main results are as follows:

Theorem 1.1Letbe the 3-rarefaction wave defined by (1.6)-(1.7)with one side being a vacuum state.Then there exists a constantε0>0 small enough such that,for anyε ∈(0,ε0),we can construct a family of global smooth solutions(ρε,mε,:=ρεθε,Zε)(t,x)to the reacting mixture equations (1.1) which satisfies that

(i)

(ii)as the coefficientεtends to zero,(ρε,mε,,Zε)(t,x)can converge topointwise except at(0,0).Furthermore,for any given positive consatntl,there exists a constantCl>0,independent ofε,such that

where the positive constantsa,b,canddare given by

The rest of this paper is organized as follows: in Section 2,we construct a smooth 3-rarefaction Navier-Stokes equation which approximates the cut-offrarefaction wave based on the inviscid Burgers’equation.In Section 3,we will list and prove a series ofa prioriestimates and assumptions.Finally,the proof of Theorem 1.1 will be given in Section 4.

NotationsWe denoteC(I;Hp(?)) as the space of continuous functions on the internalIwith values inHp(?),andL2(I;Hp(?)) as the space ofL2-functions on the internalIwith values inHp(?).For simplicity,we denote that‖·||=||·‖L2(R)and‖·‖k=‖·‖Hk(R).

2 Rarefaction Wave

For the inviscid Burgers’ equation,the Riemann problem is as follows:

IfW-

However,we know that the solutionWr()is continuous,but not smooth.In order to meet the required smoothness,inspired by Xin[36],we can constrct the smooth approximate rarefaction wave(t,x) with the solution to following Burgers’ equation

Hereδ>0 is a small parameter to be determined.We chooseδ=εain (3.6) withagiven by(1.8),so the solution(t,x) to the problem (2.2) is given by

Lemma 2.1(see [16,36]) Problem (2.2) has a unique global smooth solution(t,x)for eachδ>0 such that

1.W-<(t,x)0,forx ∈R,δ>0;

2.for anyt>0,δ>0 andp ∈[1,∞],the following inequalities hold:

3.there exists a constantδ0∈(0,1) such that,forδ ∈(0,δ0],t>0,

Due to the appearence of a vacuum,we cannot obtain a positive lower bound ofρ.To overcome this difficulty,as in [16,22],for anyρ=μ>0,we may find a state (ρμ,uμ,θμ)=(μ,uμ,μγ-1) on the 3-rarefaction wave curve.Here,uμcan be computed bylogρ++logθ+,due to the fact that the 3-Riemann invariants (1.5) are constants along the 3-rarefaction wave curve.Then we remove the part between the vacuum and the above state,and define the rest of the 3-rarefaction wave curve as a new 3-rarefaction wave(ξ) which can be expressed as

Accordingly,the momentum and total interal energy can be defined byandrespectively.

The distance between the new 3-rarefaction wave and the original one can be contolled by the non zero constantμ.

Lemma 2.2([22]) There exists a constantμ0∈(0,1) such that,forμ∈(0,μ0],t>0,

From Lemma 2.2,we know that the new 3-rarefactionconverges to the original 3-rarefaction wavein the sup-norm with the convergence rateμasμtends to zero.

Thanks to the above preparations,we can construct the approximate rarefaction wave solution(t,x)to the cut-off3-rarefaction wavefor the compressible Euler equation (1.3) by

Lemma 2.3(see [16,22,23]) The approximate rarefaction wave solutiondefined in (2.5) satisfies that:

2.for anyt>0,δ>0 andp ∈[1,∞],there exists a positive constnt C such that

3.there exists a constantδ0∈(0,1) such that,forδ ∈(0,δ0],t>0,

3 A priori Estimates

3.1 Reformulation of the Problem

We supplement equations (1.1) with the initial data

and treat the global smooth solution(ρε,uε,θε,Zε)of system(1.1)and(3.1)as the perturbation of the approximate rarefaction waveFor convenience,we establish a scale for the dependent variants

Then,if we establish the perturbation

the systems (1.1) and (3.1) can be rewritten in form of (φ,ψ,χ,Z)(τ,y) as

First,we define the functional space

with 0<τ(ε)≤∞.

We then get the global existence of solutions and some estimates.

Theorem 3.1There exist positive constantsε1andCindependent ofεsuch that,if 0<ε<ε1,the Cauchy problem (3.3) admits a unique global-in-time solution (φ,ψ,χ,Z)∈X(0,∞) satisfying that

whereais given by (1.8).Consequently,

By a standard process,we get the local existence of the solutions,and we assume that[0,τ1(?)]is the maximum existence time of the solution.In order to get the estimates,we need positive bounds for bothρandθ.To this end,we makea prioriassumptions in the following:

Here the positive constant a is given by (1.8).We set that

Indeed,ifε ?1,it is easy to see that

Next,we obtain somea prioriestimates.

Proposition 3.2(a prioriestimates) Letγ>1,and let (φ,ψ,χ,Z)∈X(0,τ1(ε)) be a solution to system(3.3),whereτ1(ε)is the maximum existence time of the solution satisfying thea prioriassumptions of (3.5).Then there exists a positive constantε2such that,if 0<ε ≤ε2,then it holds that

Consequently,

whereais given by (1.8) and the constantCis independent ofεandτ1(ε).

Now we show that the solution exists globally.

Proof of Theorem 3.1We observe that in the maximum time interval [0,τ1(?)],the estimates (3.10),(3.12),(3.13) are indeed smaller than thea prioriassumption (3.5).Hence,we conclude thatτ1(?)=∞.In fact,ifτ1(?)<∞,we may takeτ1(?) as a new initial time.According to local existence result and the continuity argument,there existsτ2(?)>τ1(?) such that there exist solutions on [0,τ2(?)]which satisfy the assumption (3.5).Thus contradicts the assumption that [0,τ1(?)]is the maximum time interval,so we can extend the local solution to the global solution in [0,∞]for small but fixed?.

The proof of Proposition 3.2 is divided into the following lemmas:

3.2 Basic Energy Estimates

Lemma 3.3Under the conditions of Proposition 3.2,for 0≤τ ≤τ(ε),we have that

ProofFirst,we define the relative entropy-entropy flux pair (η,q) as

with Φ(η)=η-lnη-1.

Direct computation yields that

withε>0 to be determined.Then we have,from [22],that

Integrating (3.15) over R×[0,τ]and using (3.6) yields that

By Sobolev’s inequality and Lemma 2.3,forIi(i=1,2,3,4),we have that

where we have used the fact that

ForI2,we can obtain,by using Cauchy’s inequality and Sobolev’s inequality,that

where we have used the fact that

Finally,forI4,we have that

Substituting (3.17),(3.18),(3.19) and (3.20) into (3.16),we can obtain (3.14).This completes the proof of Lemma 3.3.

3.3 Estimates of Higher Order Derivatives

In this subsection,we deduce the higher order estimates.

First,we obtain the estimates ofφyandψy.The proof is the same as in [7],so we omit it here.

Lemma 3.4Under the conditions of Proposition 3.2,for 0≤τ ≤τ(ε),we have that

Lemma 3.5Under the conditions of Proposition 3.2,for 0≤τ ≤τ(ε),we have that

Next,we estimate supτ‖Zy‖.

Lemma 3.6Under the conditions of Proposition 3.2,for 0≤τ ≤τ(ε),we have that

ProofMultiplying (3.3)3yields that

Then integrating the above over R×[0,τ],it holds that

For these terms on the right-hand side of (3.22),we have that

where we have used the fact that

where we have used the fact that

where we have used the fact that

where we have used the fact that

Then,plugging (3.23),(3.25),(3.27) and (3.28) into (3.22) and using Lemma 3.3.Lemma 3.6 is proven.

Finally,the estimates of supτ‖χy‖are as follows:

Lemma 3.7Under the conditions of Proposition 3.2,for 0≤τ ≤τ(ε),we have that

ProofMultiplying (3.3)4yields that

Then,integrating the last equation over R×[0,τ],it holds that

By Sobolev’s inequality,we find that

By virtue of (3.3),(3.7) and Lemma 2.3,we have that

Then,by Cauchy’s inequality,we find that

Now,we sestimateIi(i=9,···,16) as follows:

where we have used the fact that

where we have used the fact that

where we have used the fact that

where we have used the fact that

Substituting (3.35)–(3.38),(3.39),(3.41),(3.43) and (3.45) into (3.34) and combining (3.34)with (3.33),we can deduce (3.30).

Therefore,thanks to Lemmas 3.3–3.6,we obtain(3.9).It follows from(3.9)that if 1<γ ≤2,then

Hence,by the above inequalities,(3.5) holds ifε ?1.

4 Proof of Theorem 1.1

Here we know thata,b,canddare defined as in (1.8)–(1.9).It holds that,for any given positive constantl,there exists a positive constantClthat is independent ofεsuch that

Similarly,we have that

This completes the proof of Theorem 1.1.

Conflict of InterestThe authors declare no conflict of interest.