Jincheng Gao,Boling Guo

(Institute of Applied Physics and Computational Math.,100088,Beijing,PR China)

Xiaoyu Xi?

(Graduate School of China Academy of Engineering Physics,100088,Beijing,PR China)

GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR FOR THE STRONG SOLUTIONS IN H2TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS?

Jincheng Gao,Boling Guo

(Institute of Applied Physics and Computational Math.,100088,Beijing,PR China)

Xiaoyu Xi?

(Graduate School of China Academy of Engineering Physics,100088,Beijing,PR China)

In this paper,we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in threedimensional whole space.The global existence of strong solutions is obtained by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H2-framework.If the initial datas in L1-norm are finite additionally,the optimal time decay rates of strong solutions are established.With the help of Fourier splitting method,one also establishes optimal time decay rates for the higher order spatial derivatives of director.

compressible nematic liquid crystal flows;global solution; Green function;long-time behavior

2010 Mathematics Subject Classification 35Q35;35B40;76A15

Ann.of Appl.Math.

32:4(2016),331-356

1　Introduction

In this paper,we investigate the motion of compressible nematic liquid crystal flows,which are governed by the following simplified version of the Ericksen-Leslie equations

where ρ,u and d stand for the density,velocity and macroscopic average of the nematic liquid crystal orientation field respectively.The pressure P(ρ)is a smoothfunction in a neighborhood of 1 with P′(1)=1.The constantsμand ν are shear viscosity and the bulk viscosity coefficients of the fluid respectively,that satisfy the physical assumptions

The positive constants γ and θ present the competition between the kinetic energy and the potential energy,and the microscopic elastic relaxation time for the molecular orientation field,respectively.For simplicity,we set the constants γ and θ to be 1. The symbol?denotes the Kronecker tensor product such that u?u=(uiuj)1≤i,j≤3. To complete system(1.1),the initial data are given by

Furthermore,as the space variable tends to infinity,we assume

where w0is a fixed unit constant vector.The system is a coupling between the compressible Navier-Stokes equations and a transported heat flow of harmonic maps into S2.Generally speaking,we can obtain any better results for system(1.1)than those for the compressible Navier-Stokes equations.

The hydrodynamic theory of liquid crystals in the nematic case has been established by Ericksen[1]and Leslie[2]during the period of 1958 through 1968.Since then,the mathematical theory is still progressing and the study of the full Ericksen-Leslie model presents relevant mathematical difficulties.The pioneering work comes from[3-6].For example,Lin and Liu[5]obtained the global weak and smooth solutions for the Ginzburg-Landau approximation to relax the nonlinear constraint d∈S2.They also discussed the uniqueness and some stability properties of the system.Later,the decay rates for this approximate system were given by Wu[7] in a bounded domain.On the other hand,Dai et al.[8],Dai and Schonbek[9]established the time decay rates for the Cauchy problem respectively.More precisely, Dai and Schonbek[9]obtained the global existence of solutions in the Sobolev space HN(?3)×HN+1(?3)(N≥1)only requiring the smallness of, where w0is a fixed unit constant vector.If the initial data in L1-norm are finitely additionally,they also established the following time decay rates

for k=0,1,2,···,N.Recently,Liu and Zhang[10],for the density-dependent model,obtained the global weak solutions in dimension three with the initial density ρ0∈L2,which was improved by Jiang and Tan[11]for the caseUnder the constraint d∈S2,Wen and Ding[12]established the local existence ofthe strong solutions and obtained the global solutions under the assumptions of small energy and positive initial density,which was improved by Li[13]to be of vacuum.Later,Hong[14]and Lin,Lin and Wang[15]showed independently the global existence of weak solutions in two-dimensional space.Recently,Wang [16]established a global well-posedness theory for rough initial data provided that∥u0∥BMO?1+[d0]BMO≤ε0for some ε0＞0.Under this condition,Du and Wang[17] obtained arbitrary space-time regularity for the Koch and Tataru type solution (u,d).As a corollary,they also got the decay rates.For more results,the readers can refer to[18-22]and the references therein.

Considering the compressible nematic liquid crystal flows(1.1),Ding,Lin,Wang and Wen[23]gained both the existence and uniqueness of global strong solutions for one dimensional space.And this result about the classical solutions was improved by Ding,Wang and Wen[24]by generalizing the fluids to be of vacuum.For the case of multi-dimensional space,Jiang,Jiang and Wang[25]established the global existence of weak solutions for the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity if some component of initial direction field is small.Recently,Lin,Lai and Wang[26]established the existence of global weak solutions in three-dimensional space,provided the initial orientational director field d0lies in the hemisphere S+2.Local existence of unique strong solutions was proved provided that the initial datas were sufficiently regular and satisfied a natural compatibility condition in a recent work[27].Some blow-up criterions that were derived for the possible breakdown of such local strong solutions at finite time could be found in[28–30].The local existence and uniqueness of classical solutions to(1.1)were established by Ma in[31].On one hand,Hu and Wu[32]obtained the existence and uniqueness of global strong solutions in critical Besov spaces provided that the initial data were close to an equilibrium statewith a constant vector;on the other hand,Gao et al.[30]attained the global small classical solution in Sobolev spaces Hm(m≥3)and established decay rates for the compressible nematic liquid crystal flows(1.1).For more results,the readers can refer to[34]for some recent developments of analysis for hydrodynamic flow of nematic liquid crystal flows and references therein.

Recently,Wang and Tan[35]established the global existence of strong solutions and built the time decay rates for the compressible Navier-Stokes equations in H2-framework(See Matsumura and Nishida[36]in H3-framework).Precisely,if small initial perturbation belongs to H2and initial perturbation in L1-norm is finite,they built optimal time decay rates as follows

where k=0,1.This framework of time convergence rates for compressible flows has been applied to other compressible models,refer to[37-39].

In this paper,motivated by the work[35],we hope to establish the global existence and time decay rates of strong solutions for the compressible nematic liquid crystal flows under the H2-framework.First,we construct the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state(1,0,w0)(w0is a fixed unit constant vector)in H2-framework.Second,if the initial data in L1-norm are finite additionally,the optimal time decay rates of strong solutions are established by the method of Green function.Precisely,we obtain the following time decay rates for all t≥0

where k=0,1.Although angular momentum equations(1.1)3are nonlinear parabolic equations,we hope to establish optimal time decay rates for higher order spatial derivatives of director under the condition of small initial perturbation.Motivated by Lemma 3.2,we move the nonlinear terms to the right hand side of(1.1)3and deal with the nonlinear terms as external force with the property on fast time decay rates.Then,the optimal time decay rates for higher order spatial derivatives of director are built with the help of Fourier splitting method by Schonbek[40].Finally,we also study the decay rates for the time derivatives of velocity and the mixed space-time derivatives of density and director.

Notation In this paper,we use Hs(?3)(s∈?)to denote the usual Sobolev spaces with the norm∥·∥Hs and Lp(?3)(1≤p≤∞)to denote the usual Lpspaces with the norm∥·∥Lp.The symbol?lwith an integer l≥0 stands for the usual any spatial derivatives of order l.When l is not a positive integer,?lstands for Λldefined by,whereis the usual Fourier transform operatoris its inverse.We will employ the notation a?b to mean that a≤Cb for a universal constant C＞0 independent of time t.a≈b means a?b and b?a.For simplicity,we write∥(A,B)∥X:=∥A∥X+∥B∥Xand

Now,we establish the first result concerning the global existence of solutions for the compressible nematic liquid crystal flows(1.1)-(1.3).

Theorem 1.1 Assume that the initial data(ρ0?1,u0,?d0)∈H2,|d0(x)|=1 in ?3and there exists a small constant δ0＞0 such that

then problem(1.1)-(1.3)admits a unique global solution(ρ,u,d)satisfying for all t≥0,

After obtaining the global existence of strong solutions at hand,we investigate the long-time behavior for the density,velocity and direction field.

Theorem 1.2 Under the assumptions in Theorem 1.1,suppose the initial data∥d0?w0∥L2and∥(ρ0?1,u0,d0?w0)∥L1are finite additionally,then the solution (ρ,u,d)obtained in Theorem 1.1 satisfies for all t≥0,

where k=0,1,and l=0,1,2,3.

Remark 1.1For any 2≤p≤6,by virtue of Theorem 1.2 and the Sobolev interpolation inequality,we also obtain the following time decay rates:

where k=0,1,2.Furthermore,in the same manner,we also have

where k=0,1.

Remark 1.2Under the assumption of finiteness of∥d0?w0∥L2 in Theorem 1.2,one can obtain the rate of director d(x,t)converging to the constant equilibrium state w0in L∞(?3)-norm.

Finally,we also study the convergence rates for time derivatives of velocity and mixed space-time derivatives of density and director.

Theorem 1.3Under the assumptions in Theorem 1.2,the global solution (ρ,u,d)of problem(1.1)-(1.3)has the following time decay rates for all t≥0,

where k=0,1.

This paper is organized as follows.In Section 2,we establish some energy estimates that will play an important role for us to construct the global existence of strong solutions.Then,we close the estimates by the standard continuity argument and the global existence of strong solutions follows immediately.In Section 3,webuild the time decay rates by taking the method of Green function and establish optimal time decay rates for the higher order spatial derivatives of director.Finally, we also study the decay rates for the time derivatives of velocity and the mixed space-time derivatives of density and director.

2　Proof of Theorem 1.1

In this section,we construct the global existence of strong solutions for the compressible nematic liquid crystal flows(1.1)-(1.3).By a classical argument(see [36]),the global existence of solutions are obtained by combining the local existence result with a priori estimates.Since the local existence and uniqueness of strong solutions were established by Huang et al.[27],the global solutions follow in a standard continuity argument after we establish(1.5)a priori.

2.1Energy estimates

Denoting ?=ρ?1 and n=d?w0,we rewrite(1.1)in the perturbation form as

Here Si(i=1,2,3)are defined as

where the three nonlinear functions of ? are defined by

The associated initial condition is given by

Assume there exists a small positive constant δ satisfying the following estimate

for all t∈[0,T].By virtue of(2.5)and Sobolev inequality,it is easy to get

Hence,we immediately have

which can be used frequently to derive a priori estimates.The following analytic tool has been proved in Wang and Tan[41].For simplicity,we only state the results here and omit the proof for brevity.

Lemma 2.1 Let 2≤p≤∞and 0≤m,α≤l;when p=∞we require further

that m≤α+1 and l≥α+2.Then we have that for any

where 0≤θ≤1 and α satisfy

Remark 2.1 If∥f∥H2≤M,then according to Lemma 2.1 we obtain

for any α∈[0,2].Hence,under assumption(2.5),it is easy to obtain

for any α∈[0,2].

First of all,we will derive the following energy estimates.

Lemma 2.2 Under condition(2.5),then for k=0,1,we have

Proof Taking k-th spatial derivatives to(2.1)1and(2.1)2respectively,multiplying the resulting identities by?k? and?ku respectively and integrating over ?3(by parts),it is easy to obtain

Taking(k+1)-th spatial derivatives to(2.1)3,multiplying the resulting identities?k+1n and integrating over ?3(by parts),we have

Ading(2.8)to(2.9),it follows immediately that

For the case k=0,the differential identity(2.10)has the following form

Applying the H?lder,Sobolev and Young inequalities,it is easy to obtain

Integrating by parts and applying(2.6),H?lder,Sobolev and Young inequalities,it arrives at directly

Hence,with the help of(2.6),H?lder,Sobolev and Young inequalities,we deduce

By virtue of|d|=1(that is,|n+w0|=1),it follows immediately from the H?lder and Sobolev inequalities that

Substituting(2.12),(2.14)and(2.15)into(2.11)completes the proof of(2.7)for the case of k=0.Now,we turn to give the proof of(2.7)for the case of k=1.Indeed, taking k=1 in(2.10)and integrating by part yield

Applying H?lder,Sobolev and Young inequalities,we obtain

Similarly,it is easy to deduce

and

Substituting(2.17)-(2.19)into(2.16),then we complete the proof of(2.7)for the case of k=1.The proof is completed.

Next,we derive the second type of energy estimates involving the higher order spatial derivatives of ? and u.

Lemma 2.3 Under condition(2.5),then we have

Proof Taking 2-th spatial derivatives to(2.1)1and(2.1)2respectively,multiplying the resulting identities by?2? and?2u respectively and integrating over ?3(by parts),we obtain

Applying H?lder,Sobolev and Young inequalities,it is easy to obtain

Integrating by part and applying H?lder,Sobolev and Young inequalities,it arrives at

The combination of(2.22)and(2.23)gives rise to

Now,we turn to give the estimate for the second term on the right hand side of (2.21).First of all,by virtue of H?lder and Sobolev inequalities,we have

In view of(2.6),H?lder and Sobolev inequalities,we have

and

Similarly,it is easy to deduce

Combining(2.25)-(2.27)with(2.28),we deduce

Inserting(2.24)and(2.29)into(2.21),it arrives at immediately

Taking 3-th spatial derivatives to(2.1)3,multiplying the resulting identities by?3n and integrating over ?3(by parts),we obtain

The application of H?lder,Sobolev and Young inequalities,it is easy to deduce

Substituting(2.32)into(2.31),we have

The combination of(2.30)and(2.33)completes the proof of lemma.

Finally,we will use equations(2.1)to recover the dissipation estimate for ?.

Lemma 2.4 Under condition(2.5),then for k=0,1,we have

Proof Taking k-th spatial derivatives to the second equation of(2.1),multiplying by?k+1? and integrating over ?3,then we obtain

In order to deal with∫?kut·?k+1?d x,following the idea in Guo and Wang[42], we turn the time derivatives of velocity to the density.Then,applying the mass equation(2.1)1,we can transform time derivatives to the spatial derivatives,that is,

Substituting(2.36)into(2.35),it is easy to deduce

For the case k=0,applying H?lder,Sobolev and Young inequalities,we obtain

By virtue of H?lder inequality and(2.5),it is easy to deduce

and

The combination of(2.38),(2.39)and(2.40)complete the proof of(2.34)for the case of k=0.As for the case k=1,applying H?lder,Sobolev and Young inequalities, we deduce

With the help of H?lder inequality and Lemma 2.3,it arrives at

and

The combination of(2.41),(2.42)and(2.43)gives rise to the proof of(2.34)for the case of k=1.The proof is completed.

2.2Global existence of solutions

In this subsection,we shall combine the energy estimates derived in the previous section to prove the global existence of strong solutions in Theorem 1.1.Summing up(2.7)from k=l(l=0,1)to k=1,we obtain

which,together with(2.20),arrives at

On the other hand,summing(2.34)from k=l(l=0,1)to k=1,we obtain immediately

Multiplying(2.45)by 2δC1/C2and adding the resulting inequality to(2.44),it arrives at

By virtue of the smallness of δ,it is easy to obtain

Choosing l=0 in(2.46)and integrating over[0,t]yield

Since∥(?,u,?n)(t)∥H2is a continuous function with respect to time(see[27]),there exists a small and positive constant T0such that

Choosing

this,together with(2.49),gives directly

Then,applying estimate(2.48),it is easy to deduce

Thus,problem(2.1)-(2.4)with the initial data(?,u,?n)(x,T0)admits a unique solution on[T0,2T0]×?3satisfying the estimate

which,together with(2.48),yields directly

Thus,we can continue the same process for 0≤t≤nT0(n=1,2,···)and finally get a global solution on[0,∞)×?3.The uniqueness of global strong solutions follows immediately from the uniqueness of local existence of solutions.Choosing l=0 in(2.46),integrating over[0,t]and applying the equivalent relation of(2.47), we obtain

which completes the proof of Theorem 1.1.

3　Proof of Theorems 1.2 and 1.3

In this section,we will establish the time decay rates for the compressible nematic liquid crystal flows(1.1)-(1.3).First of all,the decay rates are built by the method of the Green function.Secondly,motivated by Lemma 3.2,we enhance the time decay rates for the higher order derivatives of director.Finally,we also establish the convergence rates for the time derivatives of density,velocity and director.

3.1Decay rates for the nonlinear systems

First of all,let us to consider the following linearized systems

with the initial data

Obviously,the solution(?,u,n)for the linear problem(3.1)-(3.2)can be expressed as

Here G(t):=G(x,t)is the Green matrix for system(3.1)and the exact expression of the Fourier transform b G(ξ,t)of Green function G(x,t)as

where

Since systems(3.1)is an independent coupling of the classical linearized Navier-Stokes equation and heat equation,the representation of Green function b G(ξ,t)is easy to be verified.Furthermore,we have the following decay rates for systems (3.1)-(3.2),refer to[33,43].

Proposition 3.1Assume that(?,u,n)is a solution of the linearized compressible nematic liquid crystal system(3.1)-(3.2)with the initial data(?0,u0,n0)∈L1∩H2,then

for 0≤k≤2.

In the sequel,we want to verify some simplified inequalities that play an important role to derive the time decay rates for the compressible nematic liquid crystal flows(2.1)-(2.4).More precisely,we have

Next,we establish decay rates for the compressible nematic liquid crystal flows (2.1)-(2.4).

Lemma 3.1 Under the assumptions in Theorem 1.2,the global solution(?,u,n) of problem(2.1)-(2.4)satisfies

for k=0,1.

Proof First of all,taking k=0 in(2.9),which together with inequality(2.15), we obtain the following inequality immediately

Taking l=1 specially in(2.46),it arrives at directly

which,together with(3.6),yields directly

With the help of Young inequality,it is easy to deduce

Ading both hand sides of(3.7)byand applying the equivalent relation (3.8),we have

In view of the Gronwall inequality,it follows immediately

In order to derive the time decay rates forwe need to control the termIn fact,by Duhamel principle,we can represent the solution for system(2.1)-(2.4)as

where we have used the fact

Inserting(3.12)into(3.10),it follows immediately

where we have used the fact

Hence,by virtue of the definition of F(t)and(3.13),it follows immediately

which,in view of the smallness of δ,gives

Therefore,we have the following time decay rates

On the other hand,by(3.4),(3.11),(3.14)and Proposition 3.1,it is easy to deduce

where we have used the fact

Hence,we have the following decay rates

Therefore,the combination of(3.14)and(3.15)completes the proof of the lemma.

3.2 Optimal decay rates for the higher order derivatives of director

In this subsection,we will enhance the time decay rates for the higher order spatial derivatives of direction field.This improvement is motivated by the following lemma.

Lemma 3.2For some smooth function F(x,t),suppose the smooth function v(x,t)is a solution of heat equation

for(x,t)∈?3×R+with the smooth initial data v(x,0)=v0(x).If the function F(x,t)and the solution v(x,t)have the time decay rates

Proof Taking(k+1)-th spatial derivatives on both hand sides of(3.16),multiplying by?k+1v and integrating over ?3,we obtain

which implies

For some constant R defined below,denoting the time sphere(see[40])

it follows immediately

or equivalently

Choosing R=k+3 and combining inequalities(3.18),(3.19)and the time decay rates(3.17),it arrives at directly

Multiplying(3.20)by(1+t)k+3and integrating over[0,t],we have

which implies the time decay rates

Therefore,we complete the proof of the lemma.

Motivated by Lemma 3.2,we will improve the time decay rates for the second and third order derivatives of director.

Lemma 3.3 Under the assumptions in Theorem 1.2,the global solution(?,u,n) for problem(2.1)-(2.4)satisfies

where k=2,3.

Proof Taking k=1 in(2.9),it follows immediately

In view of(2.19),we have

By virtue of(3.5),H?lder,Sobolev and Young inequalities,it arrives at

Inserting(3.23)and(3.24)into(3.22)and applying the smallness of ε and δ,we have

On the other hand,from inequality(2.31),we have

By virtue of H?lder,Sobolev and Young inequalities,we obtain

Following from the idea of inequality(2.32),we have

Inserting(3.27)and(3.28)into(3.26)and applying the smallness of ε and δ,it arrives at immediately

Combining(3.25)and(3.29)and applying the time decay rates(3.5),we get

Similar to the analysis of inequality(3.19),it follows immediately

and

The combination of(3.30),(3.31)and(3.32)yields directly

where have used the convergence rates(3.5).Multiplying(3.33)by(1+t)4,we obtain

Integrating(3.34)over[0,t],we have the following decay rate

On the other hand,applying the convergence rates(3.5),(3.35)and inequality(3.29), it arrives at

which,together with(3.32)and(3.35),yields

Multiplying(3.36)by(1+t)5and integrating over[0,t],it follows immediately

Therefore,we complete the proof of the lemma.

Proof of Theorem 1.2 With the help of Lemmas 3.1 and 3.3,we complete the proof of Theorem 1.2.

Remark 3.1 In order to obtain the rate of d(x,t)converging to w0,we suppose the finiteness of∥d0?w0∥L2 in Theorem 1.2 additionally.Then,the density and velocity(ρ,u)enjoy the same decay rate with the director field d(x,t)?w0.However, (ρ,u)will have the same decay rate with?(d(x,t)?w0)without the assumption of finiteness of∥d0?w0∥L2.

3.3 Decay rates for the mixed space-time derivatives of density and velocity

In this subsection,we will establish the decay rates for the time derivatives of velocity and the mixed space-time derivatives of density and director.

Lemma 3.4 Under the assumptions in Theorem 1.2,the global solution(?,u,n) of problem(2.1)-(2.4)satisfies

for k=0,1.

Proof By virtue of equation(2.1)1and the convergence rates(1.6),we have

Similarly,it follows immediately that

and

By virtue of(2.1)3,(1.6),H?lder and Sobolev inequalities,we obtain

In the same manner,it arrives at directly

Therefore,we complete the proof of the lemma.

Proof of Theorem 1.3 With the help of Lemma 3.4,we complete the proof of Theorem 1.3.

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(edited by Mengxin He)

?Manuscript August 6,2016

?.E-mail:xixiaoyu1357@126.com