Liquid Crystal Flow sw ith Regu larity in One D irection

ZHANG Zu jin

SchoolofM athematicsand Computer Sciences,Gannan NormalUniversity, Ganzhou 341000,China.

Liquid Crystal Flow sw ith Regu larity in One D irection

ZHANG Zu jin∗

SchoolofM athematicsand Computer Sciences,Gannan NormalUniversity, Ganzhou 341000,China.

Received 7 February 2014;Accep ted 9M ay 2014

.In this paper,w e consider the Cauchy p roblem for them odelof liquid crystal.We show that if the velocity field u satisfies then the solution is in factsm ooth.

Liquid crystals;regu larity criteria.

1 In troduction

In this paper,w e consider the follow ing m odel of liquid crystal in R3introduced by Lin[33]:

w here u=(u1,u2,u3)is the fluid velocity field,d=(d1,d2,d3)is the(averaged)m acroscopic/continuum m olecu le orientation,πis a scalar p ressure,u0and d0are the p rescribed initial data,

Hereand thereafter,w e use the summ ation convention that repeated ind icesare summ ed au tom atically over{1,2,3}.

The existence of a global-in-tim e w eak solution and the local unique strong solution hasbeen established by Lin and Liu[1].Bu tas for the incom p ressible Navier-Stokessystem(d is constant in(1.1)),w hether a given globalw eak solu tion is regu lar and w hether the localunique strong solu tion can exist globally are challenging open p roblem s.

M otivated by the regu larity criteria for the Navier-Stokes[2–13]and MHD equations[14–26],som e au thors considered the regu larity cond itions for(1.1),see[27–32]and references cited therein.

In this paper,w ew ou ld like to im p rove the regu larity criterion

established in[29].

Beforew e state the p recise resu lt,letus recall thew eak form u lation of(1.1).

Definition 1.1([1]).Ameasurablepair(u,d)issaid tobeaweak solution of(1.1)on[0,T]×R3, provided thefollow ing assertionshold:

Now ou rm ain resu lt reads:

Theorem 1.1.Let u0∈L2(R3)satisfy∇·u0=0,d0∈H1(R3)w ith|d0|≤1,and let themeasurable pair(u,d)bea given weak solution of(1.1)w ith initial data(u0,d0).If

then the solution is in fact strong,and thus classical.

By a strong solu tion,w em ean(u,d)satisfy

Rem ark 1.1.Notice that

the scaling d im ension of Theorem 1.1 can alm ostachieve 3/2.

2 Proof of Theorem 1.1

In this section,w e shallp rove Theorem 1.1.First,letus recall

Taking the inner p roduct of(1.1)1,(1.1)2w ith-△u,△2d in L2(R3)respectively,w e obtain

Com bining(2.1)and(2.2),w e deduce

Gathering(2.4),(2.5)and(2.6)in to(2.3),w e obtain

App lying Gronw all inequality,w e deduce that the solu tion pair(u,d)isa strong solu tion, as desired.

The p roofof Theorem 1.1 is com p leted.

Acknow ledgm en ts

Thisw ork w as partially supported by the Youth Natural Science Foundation of Jiangxi Province(20132BAB211007),theNationalNatu ralScience Foundation ofChina(11326138), and the Science Foundation of JiangxiProvincial Departm entof Education(GJJ14673).

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10.4208/jpde.v27.n3.5 Sep tem ber 2014

∗Correspond ing au thor.Emailaddress:zhangzuj in361@163.com(Z.Zhang)

AM SSub jectClassifications:35Q 35,76B03

Chinese Library Classifications: O175.28,O175.29