A FLUID-PARTICLE MODEL WITH ELECTRIC FIELDS NEAR A LOCAL MAXWELLIAN WITH RAREFACTION WAVE??

Teng Wang

(College of Applied Sciences,Beijing University of Technology,Beijing 100124,PR China)

Yi Wang?

(CEMS,HCMS,NCMIS,Academy of Math.and Systems Science,Chinese Academy of Sciences,Beijing 100190,China and School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,PR China)

Abstract The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions,Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces.Precisely,based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system,which was first developed in[16],we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.

Keywords fluid-particle model;rarefaction wave;time-asymptotic stability

1 Introduction and Main Result

This paper is devoted to the time-asymptotic behavior of solutions to a system modeling the evolution of dispersed particles in a compressible charged fluid under the effect of electrostatic potential forces and at the microscopic scale,the cloud of particles is described by a Vlasov-Fokker-Planck equation,and the charged fluid is modeled by Euler-Poisson equations(denoted by EP-VFP in abbreviation)with electric fields.The fluid-particle interactions are driven by a relaxation drag frictions force exerted by the macroscopic fluid onto the particles. Such a model was first introduced by Williams in the context of combustion theory[30],and can also be found in[3],which reads

where the time variablet∈ R+,spatial variablesx=(x1,x2,x3)∈ R3and the particle velocityv=(v1,v2,v3)∈R3.In system(1.3),f=f(t,x,v)is the distribution function of particles at timetand positionxand with microscopic velocityvandρ=ρ(t,x)＞0 andu=u(t,x)=(u1,u2,u3)(t,x)are the fluid density and velocity,respectively.Here the fluid pressurep=p(ρ)is given by the usualγ-laws

with the fluid constantA ＞0 and will be normalized to be 1 in the present paper.

Fluid-particle model arises in a lot of industrial applications. One example is the analysis of sedimentation phenomenon,with applications in medicine,chemical engineering or waste water treatment. Such systems are also used in the modeling of aerosols and sprays with applications,for instance,in the study of Diesel engines(see[5,30]). First,when the self-consistent electric field is ignored,some mathematical process has been made to the fluid-particle model(1.3).For example,some stability properties and a formal asymptotic analysis to this coupled system were first investigated by Carrillo and Goudon in[5].Then Mellet and Vasseur[26]used the relative entropy method to give a rigorously mathematical proof for the formal asymptotic analysis to the NS-VFP system in[5].Furthermore,Mellet and Vasseur[25]established the global existence of weak solutions to a system of a kinetic equation coupled with compressible Navier-Stokes equations,the fluid is assumed to be barotropic withγ-pressure law(γ ＞3/2).In[6],Chae,Kang and Lee showed the existence of the global classical solutions close to an equilibrium,and further proved that the solutions converged to the equilibrium exponentially for the NS-VFP system on three-dimensional torus.Then Li,Mu and Wang[12]established the global well-posedness of a strong solution to the 3D NS-VFP system in whole space with the small initial perturbation of some given equilibrium. Moreover, the algebraic rate of convergence of a solution toward the equilibrium state was obtained.For the corresponding E-VFP system(that is,no viscosity in Navier-Stokes equations),the existence of solutions for short time was proved by Baranger and Desvillettes in[1].Later,Duan and Liu[10]studied the global well-posedness of the Cauchy problem in perturbation framework under some additional conditions on initial data,and rates of convergence of solutions toward equilibrium,which were algebraic in the whole space and exponential on torus.

Besides,some new phenomena have been observed for these coupled fluid-particle system due to the interactions of the fluid and the particle,which are different from the pure Fokker-Planck equation and the compressible fluids.Recently,the global existence of smooth solutions and the optimal time-decay rate to E-VFP system(and also NS-VFP system)were established by Li et al.[14],in particular,they showed that the solutions behaved not only as entropy waves(also called diffusion wave),but also acoustic waves(also called generalized Huygens'wave)at low frequencies by using Green's function method. All the results mentioned above to these fluid-particle models are related to the existence of general weak solutions or small smooth solutions around constant equilibrium states.For the global existence and asymptotic behaviors of solutions near a local Maxwellian to the compressible fluid-particle system,Li,Wang and Wang[16]first showed the time-asymptotically nonlinear stability of rarefaction wave to the three-dimensional E-VFP system and NS-VFP system.As a by-product,a new two-fluid model with one fluid equipped with the isothermal pressure and the degenerate viscosity coefficients depending on the corresponding density function linearly,was derived by a new micro-macro decomposition around the local Maxwellian to the kinetic part of the coupled system,which is exactly same as the well-known Saint-Venant system for the viscous shallow water model satisfying the Bresch-Desjardins entropy relations.

Recently,compressible fluids coupled with the Poisson equation through the selfconsistent force arising either from modelings of self-gravitational viscous gaseous stars([7])or from the simulation of the motion of charged particles in semiconductor devices([22]),has been attracted a lot of attentions,since it has meaningful physical background. However,to the best of our knowledge,even though there are many studies on the Navier-Stokes-Poisson system,so far there are few mathematical results to clarify the nonlinear stability of wave patterns to the inviscid Euler-Poisson system. As pointed in[13],one of the main mathematical difficulties comes from the effect of the self-consistent force on the compressible fluid,since the force generally may not be expected to be L2integrable in space and time.Recently,Duan and Liu[9]proved the nonlinear stability of rarefaction waves for the Navier-Stokes-Poisson system in the case of single ions flow and two-fluid viscous model with the different masses.For the related kinetic equations,Li,Wang,Yang and Zhong[17]first showed the global solutions to the bipolar Vlasov-Poisson-Boltzmann system shall tend time-asymptotically to either viscous shock profile and rarefaction wave if the initial data is near the corresponding local Maxwellian related to the shock or rarefaction wave profiles by using a new micro-macro decomposition. Then Li,Wang and Wang[15]generalized the result to the rarefaction wave combined with viscous contact wave case.Recently,Duan and Liu[8]proved the stability of rarefaction wave to the bipolar VPB system with disparate masses by introducing a two-component macro-micro decomposition around local bi-Maxwellians.

This paper is aim to explore the wave phenomena of Vlasov-Fokker-Planck equation coupled with Euler-Poisson equations.Note that even though the fluid is inviscid,but the fluid-particle interaction drag terms and the kinetic Fokker-Planck particle model will prevent the singularity formation and recover the dynamical stability of rarefaction wave to the inviscid compressible fluid. Precisely,we want to prove the time-asymptotic stability of rarefaction wave to the EP-VFP system in the spatial one-dimensional regime,which reads

with the initial values and the far-field states given by asx→ ±∞

withρ±＞0 andu±=(u1±,0,0)tbeing prescribed constant states which are connected by the rarefaction wave solution to the Riemann problem of the corresponding 1D compressible Euler systems

and

with the corresponding Riemann initial values

and

Note that by the quasi-neutrality of the fluid and the particle,here we only consider the isothermal fluid case,that is,γ=1 in(1.2).For the particle,the four macroscopic(fluid)quantities:the mass densityn(t,x),the momentumnw(t,x)is defined by any solutionf(t,x,v)to the kinetic part of the system(1.3)as follows:

Now we describe the rarefaction wave solution to the Euler system(1.5)-(1.8).It is straight to calculate that each Euler system(1.5)for(ρ,u1)and(1.6)for(n,w1)have two distinct eigenvalues

with corresponding right eigenvectors

such that

in the non-vacuum region.Thus the twoi-Riemann invariants Σi(ρ,u1)and Σi(n,w1)can be defined by([28])

such that

Without loss of generality,we consider the time-asymptotic stability of planar 2-rarefaction wave to the Euler systems(1.5)-(1.7)and(1.6)-(1.8)in the present paper and the stability of 1-rarefaction wave can be proved similarly. The 2-rarefaction wave to the Euler systems(1.5),(1.7)and(1.6),(1.8)can be expressed explicitly through the Riemann solution to the following inviscid Burgers equation:

IfB-＜B+,then the above Riemann problem(1.11)admits a unique self-similar rarefaction wave fan solutionBr(t,x)=given by

Then the 2-rarefaction wave solutionsto the compressible Euler system(1.5)-(1.7)andto the compressible Euler system(1.6)-(1.8)can be defined explicitly by

As explained before,in(1.13),the velocityfor the two rarefaction waves should be same to make full use of the friction damping effect of the fluid-particle model and additionally,ρrshould be same under the influence of the self-consistent electrostatic potential force.

Next,we construct a smooth profile to the 2-rarefaction wave defined in(1.13)as in[24].The smooth rarefaction wave profile to(1.11)-(1.12)can be constructed by

whereε ＞0 is a small parameter to be determined. Note that ifB-＜B+,then problem(1.14)admits a unique classical solution(t,x)given by

Correspondingly,the smooth 2-rarefaction wave profile(,)(t,x)to compressible Euler system(1.5),(1.7)and(,)(t,x)to(1.6),(1.8)can be defined by

Now we state our result on the time-asymptotic stability of rarefaction wave to the EP-VFP system(1.3)and(1.4)as follows.

Theorem 1.1Suppose()(t,x)is the2-rarefaction wave defined in(1.16),then there exist positive constants ε0,ε ＜1and a global MaxwellianM?=M[n?,w?]with n?＞0,such that if the initial values satisfy

then EP-VFP system(1.3)with(1.4)admits a unique global smooth solution for all t∈ [0,+∞)satisfying

for some constant C uniform-in-time.Furthermore,the time-asymptotic stability of2-rarefaction wave holds:

Finally,we comment on some of the main difficulties and techniques involved in studying the problem of asymptotic behavior toward the rarefaction waves of solutions to the compressible EP-VFP system.Compared with the previous work of Li,Wang and Wang in[16],the main difficulty comes from the appearance of the selfconsistent electric field,which may not be expected to beL2integrable in space and time.It is quite non-trivial to estimate the coupling termsince the large-time behavior of Φ has a slow time-decay rate and the strength of rarefaction waves is not necessarily small.Motivated by Duan and Liu in[9]for the stability of rarefaction waves for the Cauchy problem on the two-fluid Navier-Stokes-Poisson system with different masses,this difficulty term could be controlled by the suitable weighted estimates to two momentum equations with different weights due to the good weak dissipative property from the Poisson equation(see(3.31)). Compared with[9],the main difference here is that the fluid is described by inviscid Euler equation,while the dynamical stability of rarefaction wave to the inviscid compressible fluid comes essentially from the fluid-particle interactions and the kinetic Fokker-Planck equation for particle model.

The rest part of the paper is arranged as follows.In Section 2,we first introduce a new micro-macro decomposition around the local Maxwellian to the kinetic part of the EP-VFP system(1.3)and consequently a new two-fluid model was found,which was first derived by Li,Wang and Wang in[16].Then the microscopic H-Theorem and some properties of approximate rarefaction waves are presented. In Section 3,we will prove our time-asymptotic stability of rarefaction wave to the EP-VFP system in Theorem 1 based on some uniform-in-time a priori energy estimates.

NotationsThroughout this paper,several positive generic constants are denoted byCwithout confusion.For functional spaces,Hs(R)denotes thes-th order Sobolev space with its norm

2 Preliminaries

In this section,we will first describe a new two-fluid model which was developed by Li,Wang and Wang in[16]through a new micro-macro decomposition around the local Maxwellian to the kinetic part of the coupled system.Then the microscopic H-Theorem will be introduced for completeness which is essential to carry out the estimates of microscopic part. Finally,the properties of approximate rarefaction waves will be listed.

2.1 New Micro-Macro decomposition

The local Maxwellian to the kinetic Fokker-Planck part of the EP-VFP system(1.3)M=M[n,w](t,x,v)can be defined as follows,

We introduce the micro-macro decomposition around the local Maxwellian in the following way. In fact,for any solutionf(t,x,v)to system(1.3),it can be decomposed as,

where the local MaxwellianMdefined in(2.1)is the macroscopic(fluid)component andGis the microscopic(non-fluid)component.

For some given global or local Maxwellian,the following weighted inner product inspace with respect to the Maxwellianis defined by

for any functionsg1(v),g2(v)such that the above integral is well-defined.Ifis the local MaxwellianMin(2.1),we shall use the simplified notation 〈·,·〉instead of〈·,·〉Mif without confusions and denote that.With respect to the inner product 〈·,·〉,the following four pairwise and normalized orthogonal base span the macroscopic space N

In terms of above orthogonal base,the macroscopic projectionP0to N and the microscopic projectionP1to N⊥can be defined as

In the following the superscript M inwill often be neglected if without confusions.Now equation(1.3)1can be rewritten as

whereLwis the linearized operator defined by

Based on the above preparation and decompositionf=M+G,the EP-VFP system(1.3)can be decomposed into the correspondingly fluid part system

The correspondingly non-fluid part equation

with

Remark 2.1Notice that the degenerate viscosity coefficient in(2.7)is exactly same as the well-known Saint-Venant system for the viscous shallow water model satisfying the Bresch-Desjardins entropy relations[4],which is first derived and observed in the previous work of Li,Wang and Wang[16]for the E-VFP system and NS-VFP system,the more detail of the derivation can be found in[16].

2.2 Microscopic H-Theorem

In this subsection,we list some lemmas on the dissipative properties of the linearized Fokker-Planck operator around the local Maxwellian in the weightedspace.All the proof can be found in[16],we only present the conclusion for completeness.

The dissipation effect of the linearized Fokker-Planck operatorLw?around the global MaxwellianM?=M[n?,w?]was proved in[5]as follows.

Lemma 2.1[5]There exists a positive constant σ such that for any g∈ N⊥,

whereN⊥denotes the vertical space of the macroscopic spaceN.

However,to deal with the energy estimates of microscopic componentG,we need the dissipation effect of the linearized Fokker-Planck operatorLwaround the local MaxwellianM=M[n(x,t),w(x,t)]depending on bothxandt,the following lemma was first proved in[16],we omit the proof for brevity.

Lemma 2.2[16]There exits some positive constant σ(w)such that for any function f,

which implies for any g∈ N⊥,

As a direct consequence of Lemma 2.2,the following lemma also holds.

Lemma 2.3There exist two positive constants(w?)and η0=η0(w?)suchthat if|w-w?|＜η0,we have for any g∈ N⊥

whereM?=M[n?,w?]is some global Maxwellian for given constants n?＞0and w?.

Remark 2.2In Lemma 2.3,the positive constantη0may not be sufficiently small. However,during the proof of Theorem 1.1 in the following sections,the smallness ofη0is crucially used to close the a priori assumptions(3.2).

As a corollary of Lemma 2.3 and Hlder inequality directly,it holds that:

Corollary 2.1Under the assumption in Lemma2.3,we have for any g∈N⊥

Moreover,it holds that:

Lemma 2.4[16]For each h(v),it holds

for z=t or x.

2.3 Properties of approximate rarefaction wave

In the following,we list some decay properties for the rarefaction wave.

Lemma 2.5[23,24]The Burgers equation(1.14)has a unique smooth global solution(t,x)such that

(1)B-＜(t,x)＜B+,x(t,x)＞0, for x∈R,t≥0.

(2)For any t ＞0and p∈ [1,∞],there exists a constant Cpsuch that

(3)The smooth rarefaction wave(t,x)and the self-similar rarefaction wavefanare time asymptotically equivalent,that is,

Correspondingly,we have the following properties of the 2-rarefaction wave profile(x,t)to the compressible Euler systems(1.17).

Lemma 2.5Let ?=|(ρ+-ρ-,u+-u-)|be the strength of the2-rarefactionwavedefined in(1.16),then the following properties hold:

(ii)The following estimates hold for all t ＞0and p∈ [1,∞]:

(iii)Time-asymptotically,the smooth2-rarefaction wave and the self-similar2-rarefaction wave are equivalent,that is,

3 The Proof of Stability Results in Theorem 1.1

In this section,we will use the new micro-macro decomposition developed in(2.7)and the dissipation properties of the linearized Fokker-Planck operator around the local and global Maxwellians in Lemmas 2.1-2.4 to prove our main results for the time-asymptotic stability of the rarefaction waves to the E-VFP system(1.3).First,we set the perturbation of the solution(ρ,u,f=M[n,w]+G)to the EP-VFP system(1.3)around the 2-rarefaction waveby

with the correction functiondefined as

One can refer the existence of the classical solution to the EP-VFP system(1.3)and(1.4)in short time to[1]. Therefore,to prove the global existence on the time interval[0,T]withT ＞0 being any positive time and the uniform-in-time estimates(1.19)and then Theorem 1.1,it is sufficient to close the following a-priori assumptions:

and verify the stability of rarefaction wave in(1.20),where and in the sequel?α=?tor?xandδis a small positive constant depending on the initial data but independent of the timeT.Note that the global MaxwellianM?in(3.2)is determined in Theorem 1.1 respectively.It can be seen from(3.2)and(1.18)that

By Sobolev's inequality,it holds that

with some positive constantC.Under the a priori assumption(3.2),we can prove that:

Theorem 3.1(A-priori estimates)Under the assumptions in Theorem1.1,any solution(ρ,u,f)to the EP-VFP system(1.4),(2.7),on the time interval[0,T]satisfies the following uniform-in-time a priori estimates:

Proof of Theorem 3.1Multiplying(3.48)by a large constantC,then adding(3.8)we can obtain(3.4),thus we finish the proof of Theorem 3.1.

Proof of Theorem 1.1The global-in-time existence of solution follows immediately from the local-in-time existence and the uniform-in-time a-priori estimates Theorem 3.1.Then we only need to justify the time-asymptotic stability of rarefaction wave as in(1.20).In fact,from(3.4)it holds that

and

Then(3.5)and(3.6)imply

It follows from one dimensional Sobolev's inequality that

which together with(3.4)and(3.7)yields

Thus

which verifies(1.20),hence the proof of Theorem 1.1 is completed.

In the following subsections,we will prove the a-priori estimates in Theorem 3.1 by the suitable combinations of the lower order estimates in Proposition 3.1 and the higher order estimates in Proposition 3.2.

3.1 Lower order estimates

Proposition 3.1Under the a priori assumption(3.2)and with the notation?α=?tor ?x,it holds that

ProofDefine the relative entropy-entropy fluxηandas

Then the direct computations yield that

Note that ifδis suitably small,then the condition≤δand Sobolev embedding theorem imply thatand|(u,w)|≤CwithCbeing a positive constant which only depends onρ-,u±. Therefore,the density functionρ(t,x):=(t,x)+φ(t,x)andn(t,x):=(t,x)+(t,x)satisfy that

since 0＜ ρ-≤(t,x)≤ρ+.Then there exists a positive constantsuch that

First,by Lemma 2.6 and(3.2),it follows from Sobolev's inequality that

By Cauchy's inequality,we have

Note that by(2.9),it holds that

It follows from Cauchy's inequality and Corollary 2.1 that

Similarly,it holds that

and

Combining(3.12)-(3.13)implies that

which together with Cauchy's inequality yields

where in the last inequality we have used the fact that

The most difficulty appears due to the fact that the electric field Φxis no longerL2integrable in space and time due to the structure of the Poisson equation in(2.7).It will be seen from the later proof that the new trouble term

can be controlled by taking the difference of two momentum equations for such two fluids model.Therefore,the structure of two-fluid model indeed plays a key role in the stability analysis of nontrivial rarefaction waves.

We rewrite(2.7)2and(2.7)5as

and

The right hand side of(3.19)will be estimated one by one. First,integrating by parts,one has

where we have used

By Lemma 2.6,Cauchy's inequality and(3.2),one has

By Cauchy's inequality,we have

It follows from Lemma 2.6 and Cauchy's inequality that

Then substituting(3.22)-(3.24)into(3.20)gives

By Lemma 2.6,Cauchy's inequality and(3.2),we obtain

here and in the sequence?is a small constant to be determine later. Similarly,it holds

and

Integrating by parts and(3.2)lead to

By Cauchy's inequality and(3.14),we derive

Substituting(3.25)-(3.30)into(3.19)yields

substituting(3.10),(3.11),(3.15)and(3.31)into(3.9),choosingεand?suitably small,one has

Next we want to get the estimation of φxandTo this end,by system(1.3)and(1.17),we obtain the following system for the perturbation:

Motivated by[29],we will estimate ‖φx‖from the gradient of the pressure term in the perturbation equation(3.33)2. More precisely,multiplying(3.33)2by φxand(3.33)4by,then adding the resulting equations together and integrating over R,one has

where in the above equality we have used(3.33)1,(3.33)3and the following fact

and

Note that

and the other terms on the right hand side in(3.34)can be controlled by

Then we chooseεsuitable small and obtain from(3.34)that

Multiplying(3.32)by a large constantC,and adding(3.36)together,choosing?andεsuitable small then integrating over[0,t],we derive

Then we derive the estimation of.First,it follows from(3.21)that

Next,multiplying equation(3.33)2byψ1t,(3.33)4byrespectively,then adding the resulted equations together and integrating over[0,t]×R,note that

then we have

Similar to(3.33)2and(3.33)4,we can deduce from(2.7)3and(2.7)6that

Multiplying equation(3.33)1byφt,(3.33)3by(3.40)1byψit,and(3.40)2byrespectively,then integrating over[0,t]×R and combining(3.38),(3.39)yield

Multiplying(3.37)by a large constantC,then adding(3.41)together give that

Finally,we carry out the microscopic estimates. By(2.8)and(3.1),satisfies equation

We can deduce from(2.4)that

Multiplying equation(3.43)by,and then integrating over[0,t]×R×R3yield that

By Cauchy's inequality,we have

and

which along with(3.45)implies that

Multiplying(3.42)by a large constantC,then adding(3.47)together implies(3.8),the proof of Proposition 3.1 is completed.

3.2 Higher order estimates

In this subsection,we will consider the higher order energy estimates.

Proposition 3.2Under the a priori assumption(3.2)and the notation ?α=?tor ?x,it holds that

The proof of Proposition 3.2 is divided into the following four lemmas.

Lemma 3.1Under the a priori assumption(3.2)and the notation ?α=?tor?x,it holds that

ProofWe simply use system(2.7)for(ρ,u,n,w)with diffusion terms in order to obtain the estimates ofdτrather than the system for the perturbationdue to the fact that the rarefaction wave itself has the decay-in-time～(1+t)-2,which is integrable with respect tot.Therefore,we rewrite system(2.7)as

and

and

Similar to that in[16],we multiply equation(3.52)1by,(3.52)2byu1xand(3.52)3by,i=2,3,respectively,then adding them together and integrating over R lead to

Similarly,multiplying equation(3.53)1by,(3.53)2byw1x,and(3.53)3bywix,i=2,3,respectively,then adding them together and integrating over R,we have

Adding(3.54)and(3.55)together,and it follows from(2.7)that

then we obtain

We just estimate the fifth term on the right hand side of(3.57)which is a typical triple nonlinear terms.By Lemma 2.6 and assumption(3.3),we obtain

It follows from(2.15),assumption(3.3)and Corollary 2.1 that

Similarly,it holds

By Cauchy's inequality and assumption(3.3),it holds

where in the last inequality we have used the one-dimensional Sobolev's inequality

Similarly,one has

Combining(3.59)-(3.61)and recalling(2.9)lead to

which together with the fact

According to Cauchy's inequality and(3.16),it holds

Integrating(3.57)with respect totover[0,t],and combining(3.58)and(3.65),we can deduce(3.49),thus the proof of Lemma 3.1 is completed.

Lemma 3.2Under the a priori assumption(3.2)and the notation ?α=?tor?x,it holds that

ProofFor this,we rewrite system(3.53)2as

Similar to(3.34),multiplying(3.52)2byρxxand(3.67)bynxx,then adding the resulted equations together and integrating over R lead to

By Cauchy's inequality and assumption(3.2),the right hand side of(3.68)can be controlled by

Integrating(3.68)with respect totover[0,t]and combining with(3.69),we can derive(3.66),thus the proof of Lemma 3.2 is completed.

Multiplying(3.49)by a large constantC,and adding it with(3.66)together imply

Then we will derive the estimations ofFirst,it follows from(3.21)that

which together with Cauchy's inequality gives

Next,multiplying equation(3.52)2byu1xt,(3.67)byw1xt,then adding the resulted equations together and integrating over[0,t]×R,one has

where we have used(3.71)and the following fact

Multiplying(3.70)by a large constantC,and adding(3.74)together imply

Lemma 3.3Under the a priori assumption(3.2)and the notation ?α=?tor?x,it holds that

ProofFirst,we can deduce from(3.44)thatP1(v1Mx)does not contain ?n,then applying?αto equation(2.8)give

Multiplying the above equation byand integrating over[0,t]×R×R3yield that

It follows from Cauchy's inequality and assumption(3.3)that

Similar to(3.46),one has,

By Cauchy's inequality and(3.62)and noting that,we deduce

Similarly,by(3.2)and(3.21),one has,

It follows from(3.2)that

Substituting(3.79)-(3.83)into(3.78)leads to(3.76),thus we finish the proof of Lemma 3.3.

Finally,in order to obtain the second order derivatives(with respect toxand/ort)estimates onG,and the third order spatial derivative estimates on the density(ρxx,nxx),we need to work on the original EP-VFP equations(1.3).

Lemma 3.4Under the a priori assumption(3.2)and the notation ?α=?tor?x,it holds that

ProofFirst,we rewrite(1.3)1as

Then applying?x?αto the above equation yields

Multiplying the above equation byand integrating over[0,t]×R×R3,we have

here we denote

SinceMt,Mx∈ N,it is known thatP1(?αMx)does not contain?α(n,w)x.Thus it holds that

From assumption(3.2)and(3.62)that,by Sobolev's inequality,one has

Using assumption(3.3)and Cauchy's inequality,it holds

and

Similar toJ1,we obtain

Similar toJ2andJ3,one has

Before we estimateJ7,noting that

where

which together with(3.86)implies that

By Cauchy's inequality,we have

where we have used the fact that

Combining(3.87)and(3.88),it holds that

By Cauchy's inequality and assumption(3.3),we have

Similarly,one has

Similar toJ7,we can obtain

Similar toJ8andJ9,it holds

Integrating by parts and using assumption(3.3)give

Combining the estimates forJ1-J14,we can deduce from(3.85)that

Then applying?αsystem(3.52)gives

Multiplying(3.90)1by(3.90)2byρ?αu1xand(3.90)3byρ?αuix(i=2,3),respectively,then adding the resulting equations together and integrating by parts lead to

Adding(3.89)and(3.91)together and noting that

and

thus we get(3.84),and the proof of Lemma 3.4 is completed.

Proof of Proposition 3.2First,multiplying(3.75)by a large constantC,then adding(3.76)together implies

Next,multiplying(3.84)by a large constantC,then adding(3.92)together and choosingδ,ε,η0suitable small yields(3.48),thus we can finish the proof of Proposition 3.2.