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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 271324, 4 pages
A Regularity Criterion for Compressible Nematic Liquid Crystal Flows
1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan
Received 2 September 2013; Accepted 26 September 2013
Academic Editors: Y. Liu and X. B. Pan
Copyright © 2013 Jishan Fan and Tohru Ozawa. 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.
We prove a blow-up criterion for local strong solutions to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in a bounded domain.
Let be a bounded domain with smooth boundary . We consider the following simplified version of Ericksen-Leslie system modeling the hydrodynamic flow of compressible nematic liquid crystals: Here is the density of the fluid, is the fluid velocity, represents the macroscopic average of the nematic liquid crystal orientation field, and is the pressure with positive constants and . Two real constants and are the shear viscosity and the bulk viscosity coefficients of the fluid, respectively, which are assumed to satisfy the following physical condition: Equations (1) and (2) are the well-known compressible Navier-Stokes system with the external force . Equation (3) is the well-known heat flow of harmonic map when .
Recently, Huang et al.  prove the following local-in-time well-posedness.
Based on the above proposition, Huang et al.  prove the regularity criterion to the problem (1)####^~^~^~^~^~^####x2013;(3), (5) with the boundary condition or Here, where is the unit outward normal vector to .
When , Huang and Wang  show the following regularity criterion: with and satisfying
2. Proof of Theorem 2
Since is the local strong solution, we only need to prove a priori estimates.
We decompose the velocity into two parts: , where satisfies and thus satisfies where we used to denote the material derivative of . Then, together with the standard theory and theory for elliptic systems, we obtain
By the theory of the elliptic equations, it follows from (3) that which yields
Testing (2) by and setting , we find that
Now we deal with the last term.
First, (1) implies that
This completes the proof.
This work is partially supported by NSFC (no. 11171154).
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- X. Huang and Y. Wang, “A Serrin criterion for compressible nematic liquid crystal flows,” Mathematical Methods in the Applied Sciences, vol. 36, no. 11, pp. 1363–1375, 2013.
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