The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations?

Yeping LI Wen’an YONG

1 Introduction

This paper is concerned with the isentropic compressible viscous magnetohydrodynamic(MHD for short)equations with a small Mach number(see[18–19]).These equations model the dynamics of compressible quasineutrally ionized fluids under the influence of electromagnetic fields and cover very wide applications of physical objects from liquid metals to cosmic plasmas.In a suitable nondimensional form(see,e.g.,[10]),the compressible viscous magnetohydrodynamic equations for an isentropic fluid read as

for(x,t)∈Ω×[0,+∞).Throughout this paper,Ω is assumed to be the 3-dimensional torus.Here the unknown functions are the densityρ,the velocityu∈and the magnetic fieldH∈.The pressurep=p(ρ)is a given strictly increasing smooth function ofρ>0.The constantsμandλare the shear and bulk viscosity coefficients of the flow,satisfyingμ>0 and 2μ+3λ≥0,respectively,the constantν>0 is the magnetic dif f usivity acting as a magnetic dif f usion coefficient of the magnetic field,andεis proportional to the Mach number.Note that,whenH=0,(1.1)reduces to the compressible Navier-Stokes equation.

It is well-known that the incompressible limit of compressible fluid dynamical equations is an important mathematical problem.Much ef f ort was made for the limit of the compressible Navier-Stokes equations and related models(see[1–3,6–7,14],etc.).Recently,Hu and Wang[8]discussed the convergence of weak solutions of the full compressible MHD flows(1.1)to the weak solutions of the incompressible viscous MHD equations in the whole space and the periodic domains,as the Mach number tends to zero.Jiang,Ju and Li[9]employed the modulated energy method to verify the limit of weak solutions of the compressible MHD equation(1.1)in the torus to the strong solutions of the incompressible viscous or partial viscous MHD equation(the shear viscosity coefficient is zero,but the magnetic dif f usion coefficient is a positive constant).The authors of[9]also derived the ideal incompressible MHD equation from the compressible MHD equation(1.1)in the whole space(d=2 ord=3)with general initial data in[10].That is,when the viscosities(including the shear viscosity coefficient and the magnetic dif f usion coefficient)go to zero,they proved that the weak solutions of the compressible MHD equation(1.1)converge to the smooth solutions of the ideal incompressible MHD equation.We remak that these results are all about the weak solutions.

In this paper,we analyze the incompressible limit for smooth solutions of the compressible magnetohydrodynamic equations(1.1).The result can be roughly stated as follows.Suppose that the initial data for(1.1)are smooth and have the form

withuε(x,0)andHε(x,0)solenoidal.Let[0,T0]be a(finite)time interval where the incompressible magnetohydrodynamic equations

with initial data

have a smooth solution.Then,forεsufficiently small,the compressible magnetohydrodynamic equations have a unique smooth solution defined for(x,t)∈Ω×[0,T0]and satisfying

Unlike those in[8–10],our result contains a sharp convergence rate and the existence time interval for(1.1)is optimal.Our analysis is guided by the spirit of the convergence-stability principle developed in[20–21]for singular limit problems of symmetrizable hyperbolic systems.In this approach,we will not derive anyε-uniform a priori estimate.Instead,we only need to obtain the error estimate in Theorem 2.3.Finally,we also thank the anonymous referees for telling us about the paper[11–12].Indeed,we completed our manuscript in 2012.Comparing with[11–12],we consider the convergence of solutions to be on the time interval where a smooth solution of the limit equations exists,and we also obtain the sharp convergence rate.These were pointed out by the anonymous referees.Moreover,our method here is different from that of[11–12].

The paper is organized as follows.Our main ideas and results are outlined in Section 2.All required(error)estimates are obtained in Section 3.

Notation 1.1|U|denotes some norm of a vector or matrixU.For a nonnegative integerk,Hk=Hk(Ω)denotes the usualL2-type Sobolev space of orderk.We writefor the standard norm ofHk,andWhenUis a function of another variabletas well asx∈Ω,we writeU(·,t)to recall that the norm is taken with respect tox,whiletis viewed as a parameter.In addition,we denote byC([0,T],X)(resp.L2([0,T],X))the space of continuous(resp.square integrable)functions on[0,T]with values in a Banach spaceX.

2 Main Ideas and Results

Our analysis is guided by the spirit of the convergence-stabilityprinciple developed in[20–21]for singular limit problems of symmetrizable hyperbolic systems.

To explain the main ideas,we firstly reformulate the compressible MHD equations(1.1)in terms of the pressurep,the velocityuand the magnetic fieldH.Sincep=p(ρ)is strictly increasing,it has an inverseρ=ρ(p).Set

Then the compressible MHD equation for smooth solutions is equivalent to

Moreover,we introduce

withp0>0 being constant.Then the above equation can be rewritten as

For(2.1),we have the following local existence of the classical solution of the initial value problem.

Lemma 2.1Let p=p(ρ)be a smooth function.Assume(x)∈H3.Then there exists a positive constant T0>0,such that(2.1)with initial datahas aunique classical solution(x,t),satisfyingfor all(x,t)∈×[0,T]and

The proof of Lemma 2.1 is similar to that in[16–17]for the compressible Navier-Stokes equation and the details can be found in[13].

Now we fixε∈(0,1].According to Lemma 2.1,there is a time interval[0,T],such that(2.1)with initial data(x,ε)has a unique solution(pε,uε,Hε)satisfyingεpε+p0>0 for all(x,t)∈×[0,T]and

Define

(Here the “2” can be replaced by any positive number larger than 1.)Namely,[0,Tε)is the maximal time interval ofH3-existence.Note thatTεmay tend to 0 asεgoes to 0.

In order to show that>0,we follow the convergence-stability principle[21]and seek a formal approximation of(pε,uε,Hε).To this end,we consider the initial-value problem(IVP for short)of the incompressible viscous magnetohydrodynamic equations:

Since(u0,H0)∈H4,we know from[5,19]that the following lemma holds.

Lemma 2.2There exists T0∈(0,+∞),such that the IVP(2.3)of the incompressible viscous magnetohydrodynamic equations has a unique smooth solution

satisfying

In the next section,we will prove the following theorem.

Theorem 2.1Suppose that p=p(ρ)is smooth and satisfies(ρ)>0for ρ>0,and u0,H0∈H4are both divergence-free.Then there exist constants K=K(T0)and ε0=ε0(T0),such that for all ε≤ε0,

for t∈[0,min{T0,Tε}).

Having this theorem,we slightly modify the arguments in[20]to prove the following result.

Theorem 2.2Under the conditions of Theorem2.1,there exists a constant ε0=ε0(T0),such that for all ε≤ε0,

ProofOtherwise,there is a sequence,such thatεk=0 and≤T0.Thanks to the error estimate in Theorem 2.1,there exists ak,such that(x,t)∈(?p0,5p0)for allxandt.Next we deduce from

and Lemma 2.2 thatis bounded uniformly with respect tot∈[0,).Now we could use Lemma 2.1,beginning at a timetless than,to continue this solution beyond.This contradicts the definition ofTεin(2.2).

By combining Theorems 2.1 and 2.2,we achieve our main result as follows.

Theorem 2.3Suppose that p=p(ρ)is smooth and satisfies(ρ)>0for ρ>0,and that u0,H0∈H4are both divergence-free.Denote by T0>0the life-span of the unique classical solution(u0,H0)(x,t)∈C([0,T0],H4)to the incompressible viscous magnetohydrodynamic equations(1.2)with initial data(u0,H0).If T0<∞,then,for ε sufficiently small,the compressible magnetohydrodynamic equation(1.1)with initial data

has a unique solution(ρε,uε,Hε)(x,t)satisfying

Moreover,there exists a constant K>0,independent of ε but dependent on T0,such that

In the case T0=∞,the maximal existence time Tεof(ρε,uε,Hε)tends to infinity as ε goes to zero.

Remark 2.1The initial data

can be relaxed as

without changing our arguments.

We conclude this section with the following interesting remarks,which is a by-product of our approach.

Remark 2.2The proof of Theorem 2.1 requiresT0<∞.However,when the IVP(2.3)of the incompressible viscous magnetohydrodynamic equations has a global-in-time regular solution,T0can be any positive number.Hence we have an almost global-in-time existence result for(2.1)as follows:

Remark 2.3In terms of the formal expansion,εp0,u0andH0are the zero-order profile of the solutionspε,uεandHε,respectively.Therefore,the convergence rate in(2.4)is sharp and optimal.

3 Error Estimate

In this section,we prove the error estimate in Theorem 2.1.For this purpose,we need the following classical calculus inequalities in Sobolev spaces(see[14]).

Lemma 3.1(i)For s≥2,Hs=Hsis an algebra,namely,for f,g∈Hs,it holds that fg∈Hsand

(ii)For s≥ 3,let f∈Hsand g∈Hs?1.Then for all multi-indices α with|α|≤s,it holds that[,f]g∈L2and

Here Csis a generic constant depending only on s.

We notice that,withu0,H0andp0as constructed in Lemma 2.2,

satisfies

with

From Lemma 2.2,it follows that

Here and belowCis a generic positive constant.

Set

Note thatuεis divergence-free.We deduce from(2.1)and(3.1)that

and

wheref1andf2are given by

and

respectively.Letαbe a multi-index with|α|≤ 3.Dif f erentiating the two sides of the equations in(3.3)–(3.5)withand setting

we obtain

First,taking the inner product of(3.8)withαover Ω yields

Here we use integration by parts for the termνΔα.It is easy to see that

whereδis a small positive number to be determined.

Forand other terms in the sequel,we follow[20–21]and formulate the following lemma.

Lemma 3.2Set

for t∈[0,min{T0,Tε}).Then for multi-indices βsatisfying|β|≤ 1,it holds that

ProofIt is obvious from Lemma 2.2 and the Sobolev inequality that

The other estimates can be showed similarly.This completes the proof.

For K3,we use Lemmas 3.1 and 3.2 to deduce that

Therefore,putting(3.10)–(3.12)into(3.9)gives

Next we take the inner product of(3.6)and(3.7)withand,respectively,and sum up the resultant equalities to obtain

Now we turn to estimate the Ii’s in(3.14).Using integration by parts,(2.1)1and Lemma 3.2,we deduce that

For I2,it follows from Lemmas 3.1–3.2 that

Like K3,I3can be simply estimated as

In order to treat other terms,we compute that

Thus,for I4we have

Similarly,

Finally,from the definitions off1andf2,we deduce that

Putting the above estimates into(3.14),we obtain

Combining the last inequality with(3.13),we arrive at

We integrate this inequality from 0 toTwith[0,T]?[0,min{Tε,T0})to obtain

Here we use the fact that the initial data are in equilibrium.Furthermore,we apply the Gronwall’s lemma to the last inequality to get

Sinceit follows from(3.15)that

Thus,it holds that

Applying the nonlinear Gronwall-type inequality in[20]to the last inequality yields

fort∈[0,min{T0,Tε})if we chooseεso small that

Thus,it follows from(3.16)thatD(T)≤forT∈[0,min{T0,Tε}).Finally,Theorem 2.1 is concluded from(3.15).This completes the proof.

AcknowledgementThe authors would like to express sincere thanks to the referees for the suggestive comments on this manuscript.

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