A Logarithmically Improved Blow-up Criterion for a Simplifled Ericksen-Leslie System Modeling the Liquid Crystal Flows

BAI Meng,LIU Qiaoand ZHAO Jihong

1School of Mathematics and Statistics,Zhaoqing University, Zhaoqing 526061,China.

2College of Mathematics and Computer Science,Hunan Normal University, Changsha 410081,China.

3College of Science,Northwest A&F University,Yangling 712100,China.

A Logarithmically Improved Blow-up Criterion for a Simplifled Ericksen-Leslie System Modeling the Liquid Crystal Flows

BAI Meng1,?,LIU Qiao2and ZHAO Jihong3

1School of Mathematics and Statistics,Zhaoqing University, Zhaoqing 526061,China.

2College of Mathematics and Computer Science,Hunan Normal University, Changsha 410081,China.

3College of Science,Northwest A&F University,Yangling 712100,China.

.In this paper,we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplifled 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal.The result shows that if a local smooth solution(u,d)satisfles

with 0≤r＜1 and s≥3,then the solution(u,d)can be smoothly extended beyond the time T.

AMS Subject Classiflcations:76A15,35B65,35Q35

Chinese Library Classiflcations:O175.26,O29

Ericksen-Leslie system;Navier-Stokes equations;blow-up criterion.

1 Introduction

Liquid crystal,which is a state of matter capable of flow,but its molecules may be oriented in a crystal-like way.Liquid crystal exhibits a phase of matter that has properties between those of a conventional liquid and those of a solid crystal,thus it is commonlyregarded as the fourth state of matter,different from gases,liquid,and solid.Nowadays,three main types of liquid crystals are distinguished,nematic,termed smectic and cholesteric.Among these three types of liquid crystals,the nematic phase appears to be the most common one,and is aggregations of molecules which possess same orientational order and are made of elongated,rod-like molecules.Since the nematic liquid crystal materials have remarkable applications in various technological flelds,there have been numerous attempts to formulate continuum theories in describing the behavior of liquid crystal flows,we refer the readers to Ericksen[1]and Leslie[2].

In this paper,we study the following simplifled 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal materials:

where u(x,t):R3×(0,+∞)→R3is the unknown velocity field of the flow,P(x,t):R3× (0,+∞)→R is the scalar pressure and d:R3×(0,+∞)→S2(the unit sphere in R3)is the unknown(averaged)macroscopic/continuum molecule orientationof the nematic liquid crystal flow,?·u=0 represents the incompressible condition,u0is a given initial velocity field with?·u0=0 in the distributional sense,d0:R3→S2is a given initial liquid crystal orientation field.The notation?d⊙?d denotes the n×n matrix whose(i,j)-th entry is given by?id·?jd(1≤i,j≤3),and hence

The above simplifled Ericksen-Leslie system was flrstly proposed by Lin[3]back in the late 1980s.It is a macroscopic continuum description of the time evolution of the rod-like liquid crystal materials under the influence of both the velocity fleld u and the macroscopic descriptionof the microscopic orientationconflgurations d.Mathematical analysis of the system(1.1)–(1.4)was initially studied by a series of papers by Lin and Liu[4,5]. Later on,there are considerably extensive studies devoted to global existence of weak solutions,global strong solutions with small initial data,uniqueness and regularity criteria of weak solutions,blow-up criteria of local smooth solutions and other related topics,we refer the readers to see,e.g.,[6–11]and the references therein.

Inthepresentpaper,weare interestedintheshorttime classical solutiontothesystem (1.1)–(1.4)and address a blow-up criterion that characterizes the flrst flnite singular time of the local classical solution.Since the strong solutions of the heat flow of harmonic maps,i.e.,the case u≡0 in(1.1)–(1.4),must be blowing up at flnite time(see[12]),we cannot expect that(1.1)–(1.4)has a global smooth solution with general initial data.Thelocal well-posedness for the Cauchy problem of the system(1.1)–(1.4)with sufflciently smooth initial data is more or less standard,more precisely,let?d∈S2be a given constant unit vector,if u0∈Hs(R3,R3)with?·u0=0 and d0??d∈Hs+1(R3,S2)for s≥3,then there exists 0＜T?＜+∞depending only on the initial data such that the system(1.1)–(1.4)has a unique local classical solution(u,d)satisfying(see e.g.,[11])

for all 0＜T＜T?.Notice that the global-in-time existence theory for classical solutions to the system(1.1)–(1.4)is still the challenge open problem.Hence,if we let T?be the maximum value so that(1.5)holds,then it is crucially to flnd some sufflcient conditions to characterize such a T?.When dimension n=2,Lin,Lin and Wang[9]obtained that the local smooth solution(u,d)to(1.1)–(1.4)can be continued past any flnite time T＞0 provided that there holds

Huang and Wang[7]established that if 0＜T?＜∞is the flrst flnite singular time,then

Here,ω:=?×u is the vorticity.These results have been improved in the optimal forms in terms of the maximal scaling invariant Besov spaces by the last two authors of this paper,see[13].Recently,Chen,Tan and Wu[14]proved a LPS’s criterion for the system (1.1)–(1.4).

Motivated by the above cited results for the system(1.1)–(1.4),the purpose of this paper is to establish a logarithmically improved blow-up criterion for the local smooth solutions of system(1.1)–(1.4)in terms of the homogeneous Besov spaces.Our main result is as follows:

Theorem 1.1.Let?d∈S2,u0∈Hs(R3,R3),d0??d∈Hs+1(R3,S2)with s≥3,?·u0=0,and let (u,d)be a local smooth solution on the time interval[0,T)to the system(1.1)–(1.4)with initial data(u0,d0).Suppose thatfor some 0≤r＜1.Then the solution(u,d)can be smoothly extended beyond the time T.In other words,if the solution(u,d)blows up at t=T,then

for any 0≤r＜1.

Remark 1.1.1.The assumption(1.6)can be regarded as a logarithmically improved version of the criterion

2.From Lemari′e-Rieusset[15]we see that

3.The criterion(1.6)can be replaced by the following apparently weaker condition: There exist s′≥s,0≤r＜1 and 0＜ε≤T such that

4.Since the d-equation(1.2)has L2decay property(see(2.4)below),we can replace the term‖d(·,t)‖Hs+1by‖?d(·,t)‖Hsin the criterion(1.6),the corresponding result of Theorem 1.1 still holds.

5.As a byproduct of the proof of Theorem 1.1 and the regularity theorem by[9], we can obtain a corresponding criterion in dimension two,i.e.,the criterion(1.6)can be replaced by

In fact,in our previous paper[13],we established a more general blow-up criterion for two dimension Ericksen-Leslie system.

The remaining part of this paper is devoted to proving Theorem 1.1.Throughout the paper,C denotes a constant and may change from line to line;‖·‖Xdenotes the norm of vector space(X(R3))3.

2 The proof of Theorem 1.1

In this section,we shall present the proof of Theorem 1.1.Let us flrst recall some basic deflnitions and fundamental results.

whereFandF?1denote the Fourier transform and its inverse,respectively.The distribution?jf is called the j-th dyadic block of the Littlewood-Paley decomposition of f.

In addition,for r∈R,we denote by Λrthe operator Λrf=F?1(|ξ|rFf)(for f∈S′(Rn) such that the right-hand side makes sense).Then for any 1＜p＜∞,the homogeneous Sobolev space is deflned by

It is well-known that(cf.,e.g.,[15])

The following lemma will play a crucial role in the proof of our main result.

where the constant C is independent of f and g.

Now we turn to the proof of Theorem 1.1,it sufflces to prove that the assumption(1.6) ensures the following a priori estimate:

In what follows we aim at proving that this is indeed the case.

We flrstly notice that for all smooth solutions of the system(1.1)–(1.4),one verifles the following basic energy law(see,e.g.,[4,9]):

Multiplying the d-equation(1.2)by d,integrating it over R3,and using the divergence free condition(1.3),one obtains

which combining with the condition|d|=1 yield that

Taking Λ on Eq.(1.1),multiplying the resulting equation by Λu,and integrating it over R3,one has

where the divergence free condition is employed again.Combining(2.5)and(2.6)together,it follows that

We shall estimate Ai(i=1,2,3,4,5)term by term.By using the H¨older’s inequality and the interpolation inequality,we can estimate A1as

By Lemma 2.1,we have

For A2,we have

where we have used the following Gagliardo-Nirenberg inequality

For A3and A4,we have

To estimate A5,by using the Leibniz’s rule and the condition|d|=1,we have

Inserting the estimates of Ai(i=1,2,3,4,5)into(2.7),we obtain

By the Gronwall’s inequality,we obtain that for any 0≤T1＜t＜T,

On the other hand,we know from the assumption(1.6)that for any small positive constant ε,there exists a corresponding T1=T1(ε)∈(0,T)such that

For all T1≤t＜T,let us deflne

where C0is the positive constant independent of either ε or T1,and Cεis a positive constant depending on ε which may change from line to line.

Next,we are going to derive a bound on M(t).To this end,we need to introduce the following commutator and product estimates(see[16]):

By applying Λsto(1.1)and Λs+1to(1.2),multiplying them by Λsu and Λs+1d respectively,and then integrating them over R3,we have

where we have used the following Gagliardo-Nirenberg inequalities:

For B2,applying the H¨older’s inequality and(2.11),we have

where we have used the following Gagliardo-Nirenberg inequalities:

Similarly,

where we have used the fact|d|=1.Inserting estimates of Bi(i=1,2,3,4)into(2.12),one obtains that

which together with energy inequalities(2.3)and(2.4)imply that

Acknowledgement

The authors are greatly indebted to the anonymous referee for his/her carefully reading and valuable comments and suggestions.M.Bai is supported by Foundation for Distinguished Young Talents in Higher Education of Guangdong,China(2014KQNCX223).Q. Liuispartially supportedbytheNationalNaturalScience FoundationofChina(11326155, 11401202)and the China Postdoctoral Science Foundation(2015M570053).J.Zhao is partially supported by the National Natural Science Foundation of China(11501453, 11371294)and the Fundamental Research Project of Natural Science in Shaanxi Province–Young Talent Project(2015JQ1004).

[1]Ericksen J.L.,Hydrostatic theory of liquid crystal.Arch.Rational Mech.Anal.,9(1962),371-378.

[2]Leslie F.,Theory of flow phenomenum in liquid crystals.In:The Theory of Liquid Crystals. Academic Press,London-New York,1979,1-81.

[3]Lin F.,Nonlinear theory of defects in nematic liquid crystals;phase transition and flow phenomena.Comm.Pure Appl.Math.,42(1989),789-814.

[4]Lin F.,Liu C.,Nonparabolic dissipative systems modeling the flow of liquid crystals.Comm. Pure Appl.Math.,48(1995),501-537.

[5]Lin F.,Liu C.,Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals.Disc.Contin.Dyn.Syst.,A 2(1996),1-23.

[6]Hu X.,Wang D.,Global solution to the three-dimensional incompressible flow of liquid crystals.Commun.Math.Phys.,296(2010),861-880.

[7]Huang T.,Wang C.,Blow up criterion for nematic liquid crystal flows.Comm.Partial Differ. Equ.,37(2012),875-884.

[8]Li X.,Wang D.,Global solution to the incompressible flow of liquid crystal.J.Differential Equations,252(2012),745-767.

[9]Lin F.,Lin J.and Wang C.,Liquid Crystal flow in two dimensions.Arch.Rational Mech.Anal., 197(2010),297-336.

[10]Lin F.,Wang C.,On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,Chinese Annal.Math.,31(2010),921-938.

[11]WenH.,Ding S.,Solutions ofincompressiblehydrodynamicflow ofliquidcrystals.Nonlinear Anal.RWA.,12(2011),1510-1531.

[12]Chang K.,Ding W.and Ye R.,Finite-time blow-up of the heat flow of harmonic maps from surfaces.J.Differ.Geom.36(2)(1992),507-515.

[13]Liu Q.,Zhao J.,Logarithmically improved blow-up criteria for the nematic liquid crystal flows.Nonlinear Anal.RWA.,16(2014),178-190.

[14]Chen Q.,Tan Z.and Wu G.,LPS’s criterion for incompressible nematic liquid crystal flows. Acta Mathematica Scientia,34(4)(2014),1072-1080.

[15]Lemari′e-Rieusset P.G.,Recent Developments in the Navier-Stokes Problem.Chapman and Hall/CRC,London,2002.

[16]Kato T.,Ponce G.,Commutator estimates and the Euler and Navier-Stokes equations. Commu.Pure Appl.Math.,41(1988),891-907.

?Corresponding author.Email addresses:124204602@qq.com(M.Bai),liuqiao2005@163.com(Q.Liu), jihzhao@163.com(J.H.Zhao)

Received 17 August 2015;Accepted 28 October 2015