- Research Article
- Open Access
A Regularity Criterion for the Nematic Liquid Crystal Flows
© Y. Zhou and J. Fan. 2010
- Received: 25 September 2009
- Accepted: 16 April 2010
- Published: 24 May 2010
A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.
- Natural Region
- Local Existence
- Crystal Material
- Regularity Criterion
- Part Yield
Very recently, results for the local existence of classical solutions for the problems (1.1)–(1.4) were presented in . The aim of this paper is to establish a regularity criterion for it. We will prove the following.
Then can be extended beyond .
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).
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 .
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 .
for any .
where , for , and .
Now we estimate each term as follows.
This completes the proof.
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).
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