ISRN Mathematical AnalysisVolume 2013 (2013), Article ID 271324, 4 pageshttp://dx.doi.org/10.1155/2013/271324

Research Article

A Regularity Criterion for Compressible Nematic Liquid Crystal Flows

Jishan Fan¹ and Tohru Ozawa²

¹Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China²Department 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.

Abstract.

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.

1. Introduction

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. [1] prove the following local-in-time well-posedness.

Proposition 1. Let for some and in , , and in . If, in addition, the compatibility condition holds, then there exist and a unique strong solution to the problem (1)–(5).

Based on the above proposition, Huang et al. [2] prove the regularity criterion to the problem (1)–(3), (5) with the boundary condition or Here, where is the unit outward normal vector to .

When , Huang and Wang [3] show the following regularity criterion: with and satisfying

When the term in (3) is replaced by , the problem (1)–(5) has been studied by L. M. Liu and X. G. Liu [4]; they proved the following regularity criterion:

The aim of this paper is to study the regularity criterion of local strong solutions to the problem (1)-(5). We will prove

Theorem 2. Let the assumptions in Proposition 1 hold true. If (12) holds true with , then the solution can be extended beyond .

Remark 3. Theorem 2 is also true for the boundary condition (9). But it is an open problem to prove (12) when the homogeneous Dirichlet boundary condition is replaced by

2. Proof of Theorem 2

Since is the local strong solution, we only need to prove a priori estimates.

First, testing (2) and (3) by and , respectively, and adding the resulting equations together, we see that which gives

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

Testing (3) by and using (4), (20), (3), and the identity , we derive for any , where we have used the Hölder inequality and the Gagliardo-Nirenberg inequality

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

Using (27) and (20), we have

Inserting (28) into (26) and using (20), we have

Combining (21), (25), and (29), taking small enough, and using the Gronwall inequality, we conclude that

Now by the same calculations as those in [3, 5], we prove that

This completes the proof.

Acknowledgment

This work is partially supported by NSFC (no. 11171154).

References

T. Huang, C. Y. Wang, and H. Y. Wen, “Strong solutions of the compressible nematic liquid crystal flow,” Journal of Differential Equations, vol. 252, no. 3, pp. 2222–2265, 2012.

T. Huang, C. Y. Wang, and H. Y. Wen, “Blow up criterion for compressible nematic liquid crystal flows in dimension three,” Archive for Rational Mechanics and Analysis, vol. 204, no. 1, pp. 285–311, 2012.

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.

L. M. Liu and X. G. Liu, “A blow-up criterion for strong solutions to the compressible liquid crystals system,” Chinese Annals of Mathematics A, vol. 32, no. 4, pp. 393–406, 2011.

X. Huang and J. Lin, “Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows,” Communications in Mathematical Physics, vol. 324, no. 1, pp. 147–171, 2013.