Regularity Criterion for the Nematic Liquid Crystal Flows in Terms of Velocity

Abstract

We study the regularity criterion for the 3D nematic liquid crystal flows in the framework of anisotropic Lebesgue space. More precisely, we proved some sufficient conditions in terms of velocity or the fractional derivative of velocity in one direction.

1. Introduction

This paper is devoted to the regularity criterion for the three-dimensional nematic liquid crystal flows: with initial data where is the velocity field, represents the macroscopic average of the nematic liquid crystal orientation field, and is the scalar pressure. The symbol denotes a matrix whose th entry is given by for ; here . Since the sizes of the viscosity constants do not play important roles in our proof, for simplicity, we assume all these positive constants to be one.

The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1–4]; the model (1) is a simplified system of Ericksen-Leslie model which was first introduced by Lin in [5], and one of the most significant works is given by Lin and Liu [6]; more precisely, they established global existence for weak solutions and classical solutions. Recently, Liu et al. in [7] established the regularity criterion for (1) as follows: One may refer to some interesting and important regularity criteria of nematic liquid crystal flows studied by many authors (see, e.g., [8–13] and the references therein). When is constant, the system (1) becomes the well-known Navier-Stokes equations. The regularity of solutions to the 3D NS equations has been widely investigated during the past fifty years; see, for example, [14–22] and so on. The aim of this paper is to establish a new regularity criterion by providing sufficient condition in terms of velocity or the fractional derivative of velocity in one direction in the framework of anisotropic Lebesgue space.

Throughout the paper, the norm of the Lebesgue spaces is denoted by and denoted the directional derivatives of a function by , the symbol  , , , and belongs to the permutation group . Denote

Theorem 1. Let with the initial data , and let the pair be the weak solution to the liquid crystal flows (1)-(2) on for some . If satisfies then can be extended beyond .

Theorem 2. Let with the initial data , and let the pair be the weak solution to the liquid crystal flows (1)-(2) on for some . If satisfies then can be extended beyond .

Corollary 3. Under the assumption of Theorem 2, if we fix , then the sufficient condition is that

Remark 4. Comparing with the corresponding results in [7], it is obvious that the conclusion of Corollary 3 is an improvement version of Theorem 1.1 in [7] in some sense.

2. The Proof of Theorems 1 and 2

In this section, we will prove Theorems 1 and 2. For convenience, we first recall the following three-dimensional Sobolev and Ladyzhenskaya inequalities in the whole space (see, e.g., [23–25]).

Lemma 5. Let , , and , . There hold that

Proof of Theorem 1. Suppose that is the maximal interval of the existence of the local smooth solution. If , then there is nothing to prove; on the other side, for , our strategy is to show that under the assumption (5). As a result, the interval cannot be a maximal interval of existence, which leads to a contradiction.
We multiply (1)₁ by and integrate over and, similarly, multiply (1)₂ by and integrate over and then by adding two results above and using the fact that , we obtain Here we used the facts that div  and ; here denotes the usual inner product of , which implies
Besides, we multiply (1)₂ by and integrate over and get which implies
Multiplying the first equation of (1) by and integrating over . Similarly, by taking on both sides of the second equation of (1), by multiplying the resulting equation by , by integrating over , and then by adding two results above and taking the divergence-free condition div into account, we obtain
In the following, we establish the bounds of , for the first term ; thanks to Lemma 5 and using Young’s inequality, we have For the second term , similar to estimate of , we have For the term , using Hölder’s inequality, Young’s inequality, and (13), one has Substituting the above estimates (15)–(17) into (14), we obtain Integrating (18) from 0 to , using Hölder’s inequality and Young’s inequality, one has
Finally, applying Gronwall’s inequality and using condition (5), then can be extended beyond . This completes the proof of Theorem 1.

Proof of Theorem 2. When , combining Theorem 1 and using the following imbedding theorem, one can get the conclusion that When , our strategy is to show that is a sufficient condition. We can verify that integral term satisfies the conditions of Theorem 1 with . Applying Lemma 5, Hölder’s inequality, and the interpolation theorem, one can conclude that, for , where with and we have used the fact that implies . Using Hölder’s inequality, one has where .
According to the fact that and , we have This together with Theorem 1 gives the desired result of Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the NNSFC (Grant no. 11271381), China, 973 Program (Grant no. 2011CB808002), Guangdong Provincial Culture of Seedling of China (no. 2013LYM0081), Guangdong Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University.