# A Regularity Criterion for the Nematic Liquid Crystal Flows

- Yong Zhou
^{1}Email author and - Jishan Fan
^{2, 3}

**2010**:589697

https://doi.org/10.1155/2010/589697

© Y. Zhou and J. Fan. 2010

**Received: **25 September 2009

**Accepted: **16 April 2010

**Published: **24 May 2010

## Abstract

A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.

## 1. Introduction

Very recently, results for the local existence of classical solutions for the problems (1.1)–(1.4) were presented in [3]. The aim of this paper is to establish a regularity criterion for it. We will prove the following.

Theorem 1.1.

Remark 1.2.

Equation (1.5) can be regarded as a logarithmically improved regularity criterion of the form with . Condition (1.5) only involves the velocity field , which plays a dominant role in regularity theorem. Similar phenomenon already appeared in the studies of MHD equations (see [4–6] for details).

Remark 1.3.

When in (1.1), then (1.1) and (1.2) are the well-known Navier-Stokes equations. Similar conditions to (1.5) have been established in [7–10]. But previous methods can not be used here.

Remark 1.4.

A natural region for
in (1.5) should be
, but we only can prove it for
here. We are unable to establish any other regularity criterion in terms of
or
*.*

## 2. Proof of Theorem 1.1

Since we deal with the regularity conditions of the local smooth solutions, we only need to establish the needed a priori estimates. We mainly will follow the method introduced in [9].

Now we estimate each term as follows.

This completes the proof.

## Declarations

### Acknowledgments

The authors thank the referee for his/her careful reading and helpful suggestions. This work is partially supported by Zhejiang Innovation Project (Grant no. T200905), NSF of Zhejiang (Grant no. R6090109), and NSF of China (Grant no. 10971197).

## Authors’ Affiliations

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## Copyright

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