A Regularity Criterion for Compressible Nematic Liquid Crystal Flows

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 H？lder inequality and the Gagliardo-Nirenberg