Congchong Guo

(Longyan University,Longyan,364000,Fujian,PR China)

WELL-POSEDNESS AND SPACE-TIME REGULARITY OF SOLUTIONS TO THE LIQUID CRYSTAL EQUATIONS
IN CRITICAL SPACE??

Congchong Guo?

(Longyan University,Longyan,364000,Fujian,PR China)

In this paper,we consider a hydrodynamic flow of nematic liquid crystal system.We prove the local well-posedness for the system in the critical Lebesgue space,and study the space-time regularity of the local solution.

space-time regularity;liquid crystal system;critical Sobolev space

2000 Mathematics Subject Classification 35B65

Ann.of Appl.Math.

32:4(2016),357-379

1　Introduction

In this paper,we consider the following hydrodynamic flow of nematic liquid crystal system:

which was proposed by Lin and Liu[25,26],as a simplified system of Ericksen-Leslie model.Here u is the velocity of the flow,d(·,t):?n→S2,the unit sphere in ?3, is the unit vector field to depict the macroscopic molecular orientation of nematic liquid crystal material,P is pressure.We denote by?d??d the 3×3-matrix whose (i,j)-entry is?id·?jd and 1≤i,j≤3.

The hydrodynamic theory of liquid crystal flow due to Ericksen and Leslie was developed in 1960’s[5,6,21,22].The model(1.1)is a simplified system of Ericksen-Leslie model,and it is a macroscopic continuum description of the time evolution ofmaterial under the influence of both the flow field u(x,t)and the macroscopic description of the microscopic orientation configuration d(x,t)of rod-like liquid crystal.

Many efforts on rigorous mathematical analysis of system(1.1)have been made, see[23,25-27,29]etc.Since the liquid crystal system(1.1)is a coupling system between the incompressible Navier-Stokes equations and the heat flow of harmonic maps,we shall first recall some results of Navier-Stokes equations as follows.

For the incompressible Navier-Stokes equations,in[19],Leray proved that for any finite square-integrable initial data there exists a(possibly not unique)global-in-time weak solution.Moreover,for two space dimensions case,[20]proved the uniqueness of the weak solution.Although the problems of uniqueness and regularity for n≥3 of Leray-Hopf weak solutions are still open,since the seminal work of Leray,there is an extensive literature on conditional results under various criteria.The most well-known condition is so-called Ladyzhenskaya-Prodi-Serrin condition,that is for some T＞0,u∈Lp(0,T;Lq(?n)),where the pair(p,q)satisfies

Under condition(1.2),the uniqueness of Leray-Hopf weak solutions was proved by Prodi[33]and Serrin[34],and the smoothness was obtained by Ladyzhenskaya[15]. The borderline case(p,q)=(∞,n)is much more subtle.

Subsequently,[8]proved the well-posedness for the Navier-Stokes equations in a scaling invariant spaceThe scaling invariant in the context of the Navier-Stokes equations is defined as:if a pair of functions(u(x,t),P(x,t))solves the incompressible Navier-Stokes equations,then

is also the solution of the incompressible Navier-Stokes equations with initial data (uλ(x,0),Pλ(x,0))=(λu0(λx),λ2P0(λx)).The spaces which are invariant under such a scaling are also called critical spaces.Examples of critical spaces for the Navier-Stokes in n dimensions are:

The study of the Navier-Stokes equations in critical spaces was initiated by Fujita-Kato[8,13],and continued by many authors,see[1,7,10,14,32]etc.

In 2003,Escauriaza,Seregin,and Sverak[7]obtained many perfect results,such as the backward uniqueness of the parabolic system and the regularity results for weak Leary-Hopf solutions u satisfying the additional condition u∈L∞(0,T;L3(?3)), as well as the local well-posedness in the critical Lebesgue space,which verified the borderline case of(1.2)for n=3.The results of[7]is the borderline case for theLadyzhenskaya-Prodi-Serrin condition(1.2),which implied that the bound of weak solution in L∞(0,T;L3(?3))plays a crucial role to the uniqueness of the weak solutions to the Navier-Stokes equations.And for the borderline case of(1.2)with n≥4,the results were established by Du and Dong[3].

Subsequently,the space-time regularity for those local solutions in critical Lebesgue space of the Navier-Stokes equations was presented by[2]and[10].Similarly the space-time analyticity results of the Navier-Stokes equations in other critical spaces,please see[9,11,16,30,31]etc.

Now,we turn to the liquid crystal system of(1.1).Recently,Lin,Lin,and Wang[24]studied the Dirichlet initial boundary value problem of(1.1),and proved the results that for any initial data(u0,d0)∈L2(?2)×H1(?,S2),there exists a global Leray-Hopf weak solution(u,d)that is smooth away from at most finitely many singularity times.Under the initial data(u0,d0)∈BMO?1×BMO,the local and global well-posedness were studied by Wang[35].Very recently,in[12], the authors were established some Serrin type(not in borderline case,see(1.2)) and Beal-Kato-Majida type regularity criterion for the weak solution to(1.1)in?3.In[28],Lin and Wang proved the borderline case for the Serrin type criterion which is more intrinsic and difficult.For classical solutions to the Cauchy problem in the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation,under a natural geometric angle condition,in[17],Lei, Li,and Zhang proved the global smooth solutions to a class of large initial data in energy space.After that,the existence of a pair of exact strong solutions to the 2D incompressible liquid crystal equations with finite energy was constructed by Dong and Lei[4].

Define

then we can establish the critical space for the liquid crystal equations(1.1)as(1.4). There are similar results for the liquid crystal system(1.1)in the so-called critical spaces.For example in[29]and[35]the well-posedness to system(1.1)in critical Sobolev spaceand in BMO×BMO?1were studied respectively.

Similar to the results of[7]of Navier-Stokes equations,the bound of the weak solutions(u,d)in the spacewill be crucial to determine the uniqueness of the weak solutions to the liquid crystal equations (1.1),see[28].In this paper,we shall present the well-posedness of the solutions to system(1.1)in critical Lebesgue spacesFurthermore,the space-time regularity of the solutions are also presented.

There are several ingredients in this paper.Firstly,we shall prove the local well-posedness for system(1.1)in the critical Sobolev spaces.This part have extended the corresponding results of Navier-Stokes equations to the liquid crystal system. Subsequently,we shall study the space-time regularity of the local solutions,which implies not only the smoothness of the local solution,but also the decay rate about time t.To prove our results,we need to verify the space-time regularity of the local solution in the time interval[0,T0].By the standard method,it is easy to prove our results in[0,T1]with T1?T0,and then some iteration method to verity our results always hold on[T1,T0].

Our results are stated as follows.

Theorem 1.1(Local well-posedness)Suppose that(u0,d0)is a pair of initial data of(1.1)with(u0,d0)∈Ln(?n)× ˙W1,n(?n),then there exists a constant T＞0, which depends on(u0,d0),such that system(1.1)admits a pair of unique solution (u,d)with the following properties:

Moreover,if the initial data satisfying∥u0∥Ln(?n)+∥?d0∥Ln(?n)is small enough, then we can take T=+∞.

Remark 1.1 In[28],when u∈L∞(0,T;L2(?n))∩C([0,T);Ln(?n))and d∈the Leray-Hopf type weak solution is unique on ?n×[0,T],moreover,the local solution is smooth on[0,T]×?n.

Theorem 1.2Let(u,d)be the local solution presented in Theorem 1.1 on [0,T],then for any positive integers k and m,by letting(p,q)∈[2,∞]×[n,∞]withwe have

This paper is organized as follows:In Section 2,we shall present some wellknown results for the Leray Projector operator and some estimates for linear stokes system.In Section 3,the local well-posedness of(1.1)in the critical Lebesgue spaces is proved.The space-time regularity properties of the local solutions are proved in Section 4.Throughout this paper,we sometimes use the notation A?B as an equivalent to A≤CB with a uniform constant C.The notation A≈B means that A?B and B?A.

2　Preliminaries

At the beginning,we recall some properties for the Leary projection operator P to divergence free vector fields,which is defined by its matrix valued Fourier multiplierFor any multi-indices α,this symbol satisfies Mihlin-HormanderconditionFurthermore,we have the following pointwise bound (see[18]Proposition 11.1).

Lemma 2.1Denote etΔas the heat operator,n as the space dimension andas the kernel of?k+1P etΔrespectively,then there holds

where C(k)is a constant depending only on k.

with degree k+2m,such that

Proof It can be proved by induction.

We also need the following properties of the solution to the heat equation:

Lemma 2.3 Denote φ to be the solution to the linear heat equation?tφ?Δφ=0 with the initial data φ|t=0=φ0.Then for n≥2 there hold:

(1)For s≥s1≥1,denote

Particularly,for the case n=s1≥2,s=l=n+2,we have

(2)When the initial data φ0∈Ln(?n),for any positive integers M and K we have

(3)For any positive integers m,n and p∈[n+2,+∞],q∈[n,n+2]satisfying the conditionwe have

where C(m,k,n)is a constant depending on m,k and n.

Proof The case(3)was proved by Dong-Du[2].For the case(1),when n=3, it was proved by Lemma 7.1 of[7].In fact(2.3)can easily be proved by using Younginequality.For the case n=s1=3,s=l=5,(2.4)is proved by[7](see Lemma 7.1).We shall prove(2.4)for arbitrary dimensions n≥2 for completeness.

Multiplying|φ|s1?2φ to the heat equation and integrating on ?n,we have

Now,we are going to prove(2.6).Let ω?be the smoother kernel and

then

For any positive integers m and k,by Lemma 2.2,we have

Recalling that φ0∈Ln(?n)and(2.10),let ?→0 and t→0,we get

The proof is completed.

Remark 2.1 Particularly,for given positive constants m and k,we can prove the following estimate just similar to(2.3):

We also recall some results for the following linear Stokes system:For given initial data u0∈Ln(?n),we have:

Proposition 2.1 For any T＞0,suppose thatand the initial data u0∈Ln(?n),then for the linear equation(2.15),there exists a uniform constant C0,such that the solution u satisfies:

Proof This Proposition comes from[7],where the authors proved it for the case n=3.For completeness,we shall give a brief proof of(2.18)-(2.19)for the general case n≥2.

Write g=|u|n/2,we have

By multiplying|u|n?2u to(2.15)and integrating by parts,we have

Then by Gronwall inequality,we get

By(2.22)and(2.23),we verify(2.18)as follows:

Similarly,(2.19)can be proved as

The proof is complete.

Lemma 2.4 For any constant k0＞0 and d0(x)∈S2,there exists a constant C(k0)such that

where dist(·,·)is the distance.

Proof This Lemma follows directly from Lemma 2.1 of[35].

3　Local Existence

We prove Theorem 1.1 by using the fixed point argument.Given any T＞0,we write

At the beginning,we set a suitable space as follows(for more details of the suitable space see[18]):

Definition 3.1 For the functions(f(x,t),g(x,t))defined on ?n×[0,T](0＜T≤∞),we say that(f(x,t),g(x,t))∈ETif there hold:

and

where

It is easy to check that both ETandare non-empty Banach spaces.

Let u(1)and d(1)be solutions to the following equations respectively,

Here and hereafter,we denote S(t)as the heat operator and P is the Leary projection operator.

By Lemma 2.3,to prove Theorem 1.1,it is sufficient to estimate(u(2),d(2)).

Proposition 3.1 There exists a constant t1,when 0＜t≤t1we have

Proof We need to prove

as well as

We shall prove(3.10)-(3.12)term by term.

When 0＜τ＜t/2,by H?lder inequality we have

For the case t/2＜τ＜t,we get

Then(3.10)comes from(3.13)-(3.15).

To verify(3.11),we begin with the term u(2)and we have

By Lemma 2.1 we have

therefore we have

Furthermore,for a uniform constant C1,from(3.16)-(3.19)we have

Similarly to the process of(3.16)-(3.21),we can get

For a uniform constant C2,from(3.22)we have

Then(3.11)follows from(3.20)and(3.22).

By Proposition 2.1,we have

where C3and C4are uniform constants.

Claim There exists a constant t1,for 0＜t＜t1there holds

Then from(3.21),(3.23)and(3.24)-(3.25),(3.12)follows immediately.

In the following,we shall verify Claim(3.26).Denote

where ω?is the usual smoother kernel.

Taking t1small enough,such that for 0＜t≤t1,we have

Recalling that ω?is the smoother kernel,we can choose the parameter ? such that

Then(3.26)follows from(3.28)-(3.30).The proof is completed.

Proposition 3.2 There exists a constant t2＞0,when 0＜t≤t2,such that

is a contraction map.More precisely, =(u0,d0),then there exists a constant t2＞0,such that for 0＜t≤t2there holds:

Proof For simplicity,we writeRecalling(3.8),we obtain

Repeating the proof as in Proposition 3.1,we have

As the proof in(3.28)-(3.30),we can take t2＞0,such that for 0＜t≤t2,we have

Then(3.32)follows from(3.35)-(3.37).The proof is completed.

Proof of Theorem 1.1 By taking

combing Lemma 2.3,Proposition 3.1 and Proposition 3.2,there exists a pair of unique solution(u,d)satisfying(1.6)-(1.7)in the time interval[0,T].To finish the proof of Theorem 1.1,we still need to verify that|d(x,t)|=1.

Following the line of[35],by Lemma 2.4 and(3.13)-(3.15),we have

Furthermore,we take the vectorwith

where we used?Q,?2Q∈the tangent plate of S2and

Then we finish the proof of Theorem 1.1.

4　Space-time Regularity of the Local Solution

In the following,we shall prove Theorem 1.2.For any positive integers M,K and(p,q)∈[2+n,∞]×[n,n+2]satisfying

it is sufficient to prove that the local solution(u,d)of(1.1)satisfies

To prove(4.2),it is sufficient to verify the special case m=0.Since when m≥1, by using the linear heat equation?tΦ?ΔΦ=F,we have

with a small modification of the following proof,and the general case m≥1 can be proved by induction.

We write

Therefore,we verify(4.2),it is sufficient to prove

To prove(4.5),we firstly give the following proposition.

Proposition 4.1Let(u,d)be a local solution on t∈[0,T]to(1.1)with the initial data u0and?d0∈Ln(?n),for any positive integers M,K and(p,q)∈[2+n,∞]×[n,n+2]satisfying(4.1),then there exists a constant 0＜δ＜T,such that

Proof We prove this proposition by fixed point argument.

Recalling Lemma 2.3,it is sufficient to estimate(u(2),d(2))with(u(2),d(2))satisfying

And we define the map T=(T1,T2)as in(3.8).

Write

we define the following space

with

Step 1 There exists a constant δ0＞0 such that

We begin the estimates with the term u(2).Taking the positive integers k≤K, we have

therefore we have

Similarly to the process of(4.11)-(4.14),we get

Next,we give an estimate forDue to Minkovski inequality,we have

From Lemma 2.2 and Young inequality

Similarly to(4.13),we have

From(4.16)-(4.18),we have

Similarly to(4.16)-(4.18),we also have

Now,we are going to estimate

From Lemma 2.2 and Young inequality

Similarly to(4.13),we have

From(4.21)-(4.23)and Young inequality,we have

Similarly to(4.21)-(4.25),we get

From(4.11)-(4.15)and(4.25)-(4.26),applying the summationwe get

Repeat the progress as in(3.26)-(3.30),we can choose a δ0＞0 small enough, such that for any t∈[0,δ0],there holds

then we finish the proof of Step 1.

Step 2 There exists a δ1＞0 such that T is a contraction map on

We choose a δ1＞0 small enough,such that for any t∈[0,δ1],there holds

Then we finish the proof of Step 2.

Taking δ=min{δ0,δ1},we conclude that there exists a unique pair of solutionBy the uniqueness of the solution,for(u,d),the local solution to (1.1),we have(u,d)=(u?,d?)in the time interval[0,δ].

We finish the proof of Proposition 4.1.

Remark 4.1If we take the initial datasmall enough,which implies the global existence,then after a slight modification of Proposition 4.1,we can prove the results of Proposition 4.1 on t∈(0,+∞).For this situation,we can get the following decay estimates immediately:

for any t＞0 and integer k≥0.

Proof of(4.5)By Proposition 4.1,it is sufficient to prove Theorem 1.2 on [δ,T]with T＜∞and δ≤T.

Denote

By the local existence in Theorem 1.1,(U,D)is the solution to(1.1)on[δ/2,T]with the initial dataDue to the results of Lin-Lin-Wang[24],we have

We can write(U,D)as

Similarly to(4.16),by using H?lder inequality and(4.25),we have

From(4.34),(4.33)and Proposition 4.1,for any integer j≥0,we have

Similarly to(4.37)-(4.38),we have

By adding(4.38)and(4.39),and using Gronwall inequality,we get

By Proposition 2.1,we have

From(4.41),recalling that(4.33)and(4.40),we have

From(4.35),(4.40)and(4.42),we have

Similarly,we can also get

Combining Proposition 4.1 and the estimates(4.40),(4.42)and(4.44),we finish the proof of(4.5).

For the complete proof of Theorem 1.2,we use the induction as the explanation at the beginning of this Section.

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(edited by Liangwei Huang)

?This work was supported by NSF of China(grant No.11471126).

?Manuscript October 17,2016

?.E-mail:guocongchong77@163.com