Blowup Criteria for Full Compressible Navier-Stokes Equations with Vacuum State?

Yongfu WANG Shan LI

1 Introduction

The motion of compressible viscous fluid with heat-conduction in R3is governed by the following full compressible Navier-Stokes equations

and the initial conditions

Here we denote by ρ,u and θ the unknown density,velocity fields,temperature of the fluid,respectively.P=Rρθ(R>0)is the pressure of the fluid.D(u)is the deformation tensor,which is described as

The shear viscosity coefficientμ and bulk viscosity coefficient λ satisfy the physical restrictions

The constant cυis the heat capacity,κ (>0)is the heat conductivity.

The following boundary conditions are considered in this paper,for some constant eρ≥0,

in weak sense.

In the absence of vacuum,the global existence of classical solutions to the system(1.1)has been established by Mastumura and Nishida in[21],when the initial data is close to an non-vacuum equilibrium in some Sobolev spaces Hs.Later,in[13],Ho ffobtained the global existence of weak solutions for discontinuous initial data when initial density and temperature are strictly positive.

In case of that the initial vacuum is allowed,this problem becomes much more complicated to the system(1.1).Feireisl[10] first proved the global existence of the variational weak solutions to the full compressible Navier-Stokes equations in dimension N≥2.Especially,Lions[20]proved the global existence of weak solution to isentropic compressible Navier-Stokes system.However,the global existence or finite-time blowup of strong solution is still an open problem,and only local existence results have been obtained for sufficiently regular data with some compatibility conditions.For details,in[4]Cho and Kim showed the local existence of the strong solution to 3D compressible Navier-Stokes equations(see also in[11–12,19–20]for isentropic flows).Recently,the global existence and uniqueness of classical solutions to the Cauchy problem in three spatial dimensions with smooth initial data with small energy is obtained by Huang,Li and Xin[17].

Meanwhile,the regularity and uniqueness of weak solution to 3D compressible Navier-Stokes equations with large data remains open.In the signi ficant work[27],Xin showed that the classical solutions will blow up in finite time when initial density has compact support.Therefore,we would not expect higher regularity of Lions’weak solutions in general.In additional,in[28]Xin and Yan showed that any classical solutions of viscous compressible fluids without heat-conduction will blow up in finite time,as long as the initial data has an isolated mass group.

Hence,it is natural to study the blowup mechanism and the possible singularity of the smooth solutions.We would like to mention the two well-known blowup criteria,Beale-Kato-Majda criterion in[1]for incompressible inviscid flows and Serrin-type in[23]for incompressible viscous flows.Namely,if T?<∞is the maximal time for the existence of a strong(or classical)solution,then

and

where

A natural problem is that whether the blowup criteria are valid for 3D compressible Navier-Stokes equations.Firstly,Huang,Li and Xin[16]have shown that maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong)solutions to compressible isentropic Navier-Stokes system.We would like to mention the references[15–16,24]and references therein for the blowup criteria to the 3D barotropic viscous flows.Moreover,for 3D full Navier-Stokes equations,Fan et al.in[9]established the following blowup criterion

with the additional condition(1.6)

and positive initial density.Recently,a BKM criterion

was established in[25]for 3D compressible Navier-Stokes equations with heat-conduction with the stringent condition(1.6).

Furthermore,Huang and Li in[14]removed the condition(1.6)and established the following blowup criterion

The motivations in this paper come from the following two facts.The one is that for compressible viscous flows with heat-conduction in dimension two,a blowup criterion only involving the divergence of velocity fields has been established by Wang in[26].In threedimension,by the blowup criterion(1.7),we don’t know whether the velocity gradient tensor plays an essential role in the blowup mechanism.Inspired by the results(see[26])in two dimension,we strongly expect to show the diagonal elements of the velocity gradient tensor plays a leading role in the possible singularity of solution instead of the velocity gradient tensor itself.The second fact is that for 3D incompressible viscous flows,Cao and Titi in[3]established the blowup criterion involving one entry of the velocity gradient tensor,which implies that onlyone entry of the velocitygradienttensorcan guarantee the global regularity of3D incompressible viscousflows.Hence,the major purpose of this paperis to establish a blowup criterion for 3D full compressible Navier-Stokes equations(1.1),in terms of the temperature θ and divu.In additional,this result improved the previous blowup criteria in[9,14,25],substituting?u(or D(u))by divu.

Throughout this paper,we adopt the following simpli fied notation

and the simpli fied ones for standard homogeneous and inhomogeneous Sobolev spaces

where 1≤r≤∞and k is a positive integer.

Next,we give the de finition of strong solutions as follows.

De finition 1.1(Strong Solutions)For>0,(ρ,u,θ)is called a strong solution to(1.1)in R3×(0,T),provided that for some r0∈(3,6],

(ρ,u,θ)satis fies both(1.1)almost everywhere in R3× (0,T)and initial condition(1.2)almost everywhere in R3.

The main results in this paper are stated as follows.

Theorem 1.1Suppose that the initial data(ρ0,u0,θ0)satisfy

for q0∈(3,6]and compatibility conditions

with(g1,g2) ∈ L2,and(ρ,u,θ)is the strong solution of the initial boundary value problem(1.1)–(1.2)together with(1.3).If T? < ∞ is the maximal time of existence,then

with

We would like to give some comments on our results.

Remark 1.1Theorem 1.1 shows that divu plays an important role in the mechanism of blowup for the 3D compressible viscous flows.If we compare with the 3D incompressible viscous flows,when the density and the temperature remain constants,the Leray-Hopf weak solution is the unique strong solution,provided that the pressure possesses some nice regularity(see[2]for global regularity criterion for the pressure).For the heat-conduct compressible Navier-Stokes equations,the pressure is determined by the density and temperature.In the proof of our results,some estimates for the pressure can be obtained,as long as some regularity assumptions on the divu and θ are given apriorily.Due to these facts,our results seem to be natural and reasonable.

Remark 1.2Recently,Huang et al.in[18]provided a Serrin-type blowup criterion for the Cauchy problem of system(1.1),roughly speaking,that if T?<∞is the maximal time for the existence of a strong solutions,then

where s and r are also restricted by(1.4).Similarly,the result implies that the divergence of velocity plays a key role in the blowup mechanism instead of the velocity gradient tensor.

Remark 1.3If θ≡ θ0≡ 0,Theorem 1.1 directly yields the following blowup criterion for the three-dimensional compressible Navier-Stokes equations,more precisely,if T?<∞is the maximal time for the existence of a strong solution,then

which is consistent with the corresponding result in[5].Some similar blowup criteria for the isentropic compressible magnetohydrodynamic flows in two dimensions and liquid-gas two-phase flow model have been established in the recent papers[6–8].

The remain of this paper is organized as follows.In Section 2,we will recall some elementary facts and inequalities.The proof of Theorem 1.1 will be given in Section 3.

2 Preliminaries

In this section,we first give some results for the existence of the local strong solution,which have shown in[4]for the initial-boundary value problem(1.1)–(1.3).

Lemma 2.1Assume that the initial data(ρ0,u0,θ0)satisfy(1.8)–(1.10),then there exists a positive constant T0and a unique strong solution(ρ,u,θ)to the problem(1.1),(1.2)together with(1.3)on R3×(0,T0].

Next,we recall some important inequalities,which will play an important role in the following arguments(see[10,22]for the details).

Lemma2.2For q∈ (1,∞)and r∈ (3,∞),there exists a positive constant C,such that for any f∈H1,g∈Lq∩D1,r,we have

where C depends only on q,r.

It thus follows from the momentum equations that we have the following elliptic system

where

are the effective viscous flux,the material derivative of f,and the vorticity of the velocity fields,respectively.

It follows from the standard Lp-estimate for the elliptic system(2.3)that we have the following lemmas(see[17]for details).

Lemma 2.3Let(ρ,u,θ)be a solution of(1.1),then there exists a general positive constant C depending only on λ andμsuch that for any p∈[2,6],

In order to estimate ∥?u∥L∞,we introduce the following BKM-type inequality,which can be found in[15].

Lemma 2.4For q∈(3,∞),suppose?u∈L2∩D1,q.There is a constant C depending on q,such that

3 Proof of the Main Results

In this section,we will show Theorem 1.1 by the contradiction arguments.We assume the contrary to the results of Theorem 1.1,namely,there exists a bounded positive constant M,such that

withWithout loss of generality,we assume ρe=0 in the following.

The upper bound estimate of the density ρ is standard,which comes from the estimate(3.1)and the continuity equation immediately(see[15–16]for details).

Lemma 3.1Suppose that

Then

Throughout this paper,C,Cidenote some generic constants depending only on M,μ,λ,R,κ,cυ,T? and the initial data.

Next,we will give the energy inequality as follows.

Lemma 3.2Under the assumption(3.1),it holds that for 0≤ T

ProofApplying standard maximum principle to the temperature equation in(1.1)together with θ0≥ 0(see[9–10])yields that

Denote the speci fic energy asand it follows from(1.1)that

Integrating(3.5)over R3×[0,T]yields

Next,multiplying the momentum equations by u and integrating the resulting equation in R3,yield that

Moreover,adding(3.6)to(3.7)implies the estimate(3.4)by Gronwall’s inequality and(3.2).This completes the proof ofLemma3.2.

The following lemma gives the key estimate of L∞(0,T;L2)-norm of?u.

Lemma 3.3Under the assumption(3.1),the estimate

holds for 0≤ T

ProofMultiplying the third equation in(1.1)by θ and integrating the resulting equation over R3give that

In order to estimate the last term on the right-hand side of(3.9),multiply the momentum equation by uθ and integrate the resulting equation over R3to obtain

Combining(3.9)and(3.10),after choosing ε suitably small,yields that

On the other hand,multiplying the second equation in(1.1)by utand integrating equation over R3yield

then it follows from Young’s inequality that

where we have used the de finition of the effective viscous flux F in(2.4).

For the last term in(3.12),it follows from the third equation in(1.1)and(2.4)that

According to the estimates(2.6),(3.3)and Young’s inequality,we have

Choosing ε suitably small and together with(2.6)and(3.12)yields that

Taking a constant C1>0 with

and

adding(3.11)multiplied by C1to(3.13)and(3.10),after choosing ε, δ suitably small,one has

Note that we can choose constant C1sufficiently large such that the inequality(3.14)holds.

In the following,it suffices to estimate the key terms of

In fact,multiplying the momentum equation by 4|u|2u and integrating the resulting equation over R3lead to

which implies

Then choosing the constant η suitably small such that 4μ ? η >0,and adding(3.16)multiplied byto(3.15),we have

Next,we estimate the term ofrespectively.

It follows from H¨older inequality and(2.1)that

Finally,H¨older’s inequality and Young’s inequality yield that

whereandwhich implies that

due to the interpolation inequality and(3.4).

Especially,when α=2,we have the following estimate:

In summary,we can choose2 and

Combining(3.17)–(3.20),it follows form Gronwall’s inequality and the estimates(3.1),(3.4)that

Finally,thanks to the condition(3.14),we obtain the estimate(3.4)and complete the proof of Lemma 3.3.

Lemma 3.4Suppose that the condition(3.1)holds.We obtain that

ProofApplying the operator[?t+div(u·)]to the j-th equation of the momentum equations(j=1,2,3)and integrating the resulting equations over R3,we obtain after integration by parts that

It follows from integration by parts and the continuity equation(1.1)that

By Young’s inequality,one has

where we have used the estimates(2.1),(3.3)–(3.4)and(3.8).

Furthermore,for the second term I2,integrating by parts leads to

Hence,it follows from Young’s inequality that

Similarly,

Substituting(3.24)–(3.26)into(3.23),and using the estimate(2.7)with p=4 and(3.8)yield that

On the other hand,multiplying the third equation in(1.1)by˙θ and integrating the resulting equation over R3give that

For the first term J1,integrating by parts and applying Young’s inequality,Gagliardo-Nirenberg inequality and(2.1)give that

By the standard L2-estimate of the third equation in(1.1)and H¨older inequality,one has

In fact,we have used the interpolation equality here.

Then,substituting(3.30)into(3.29)yields that

For the second term in the right hand side of(3.28),a series of direct computation yields that

Then,by the elementary inequalities and the interpolation inequality,we have

Similarly to the arguments to J2,we obtain that

Finally,it follows from(2.1)and the basic inequalities that

Substituting(3.31)–(3.34)into(3.28),and choosing ε suitably small give that for any η ∈ (0,1],

On the other hand,it follows from(2.7)that

Then,substituting(3.36)into(3.35)yields that

Hence,choosing η suitably small and adding(3.27)multiplied by C3=to(3.37),we obtain that

If follows from Gronwall’s inequality and(3.8)that

Finally,note that by(2.7)and the elementary inequalities,one has

Substituting it into(3.39)yields(3.22)and this completes the proof ofLemma3.4.

Lemma 3.5Suppose that the conditions(3.1)holds.We have

ProofFirst,it follows from the estimates(2.7),(3.8),(3.22)and(3.30)that we have the following fact

Furthermore,applying the operator ?t+div(u·)to the third equation in(1.1)and a series of direct computations give that

Then,multiplying(3.42)byafter integration by parts and using(3.8),(3.22)and(3.41)yield that

Applying Gronwall’s inequality,(3.22)and(3.41)directly gives(3.40).

Finally,the following lemma gives the higher order estimates of the solutions.

Lemma 3.6Suppose that the condition(3.1)holds.We have

ProofFirst,combining the known estimates(3.8),(3.22)and(3.40)and the inequalities(2.1)–(2.2),we have

and

Thus from(2.1)–(2.2),(2.5)–(2.6),we have

Next,for 2≤ p≤ q0(3

Then,we have

where we have used the following facts

due to(3.22),(3.44)–(3.45)and interpolation inequality.

Thus following from(2.8),(3.8)and(3.48),we have

Let p=q0,substituting(3.49)into(3.47),and using(3.46),we get

This together with Gronwall’s inequality and(3.22)gives that

Combining(3.22),(3.46)and(3.49),we have

Then,taking p=2 in(3.47),we obtain

due to(3.22)and(3.51).

Moreover,letting p=2 in(3.48)and together with(3.45),(3.52)and(3.22)yields

Hence,the estimates(3.44),(3.50),(3.52)–(3.53)imply(3.43)and we complete the proof of Lemma 3.6.

With the aid of the a priori estimates established above,we will complete the proof of Theorem 1.1.

In fact,the generic constants C in Lemmas 3.1–3.6 remain uniformly bounded for all TT?.Furthermore,the functions satisfy the conditions imposed on the initial data at thetime t=T?.In additional,we have∈C([0,T];L2),which implies

The compatibility conditions are given as follows:

where

and

It is clear that g1,g2∈ L2due to the estimates(3.22),(3.40)and(3.43).Thus,(ρ,u,θ)(x,T?)satisfy compatibility conditions(1.9)and(1.10).Therefore,the local strong solution beyond T?can be extended by taking(ρ,u,θ)(x,T?)as the initial data andLemma2.1,which contradicts to the assumption on T?.This completes the proof of Theorem 1.1.

AcknowledgementThe authors would like to thank the referees for careful reading and valuable suggestions to update our paper.

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