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.
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.
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).