- 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.
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).
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 .
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).
- de Gennes PG: The Physics of Liquid Crystals. Oxford University Press, Oxford, Mass, USA; 1974.MATHGoogle Scholar
- Lin F-H, Liu C: Nonparabolic dissipative systems modeling the flow of liquid crystals. Communications on Pure and Applied Mathematics 1995, 48(5):501–537. 10.1002/cpa.3160480503MathSciNetView ArticleMATHGoogle Scholar
- Sun H, Liu C: On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems. Series A 2009, 23(1–2):455–475.MathSciNetMATHGoogle Scholar
- He C, Xin Z: On the regularity of weak solutions to the magnetohydrodynamic equations. Journal of Differential Equations 2005, 213(2):235–254. 10.1016/j.jde.2004.07.002MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Remarks on regularities for the 3D MHD equations. Discrete and Continuous Dynamical Systems. Series A 2005, 12(5):881–886.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Regularity criteria for the generalized viscous MHD equations. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 2007, 24(3):491–505.View ArticleMathSciNetMATHGoogle Scholar
- Montgomery-Smith S: Conditions implying regularity of the three dimensional Navier-Stokes equation. Applications of Mathematics 2005, 50(5):451–464. 10.1007/s10492-005-0032-0MathSciNetView ArticleMATHGoogle Scholar
- Fan J, Gao H: Regularity conditions for the 3D Navier-Stokes equations. Quarterly of Applied Mathematics 2009, 67(1):195–199.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y, Fan J: Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. SubmittedGoogle Scholar
- Zhou Y, Gala S: Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces. Journal of Mathematical Analysis and Applications 2009, 356(2):498–501. 10.1016/j.jmaa.2009.03.038MathSciNetView ArticleMATHGoogle Scholar
- Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Communications on Pure and Applied Mathematics 1988, 41(7):891–907. 10.1002/cpa.3160410704MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.