A note on regularity criterion for 3D compressible nematic liquid crystal flows
© Chen and Fan; licensee Springer. 2012
Received: 11 November 2011
Accepted: 8 March 2012
Published: 8 March 2012
In this article, we prove a regularity criterion for the local strong solutions to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials.
Mathematics Subject Classifications (2010): 35M10; 76N10; 82D30.
(1.1) and (1.2) is the well-known compressible Navier-Stokes system with the external force -Δd·∇d. (1.3) is the well-known heat flow of harmonic map when u = 0.
Very recently, Ericksen  proved the following local-in-time well-posedness:
holds, then there exist T0 > 0 and a unique strong solution (ρ, u, d) to the problem (1.1)-(1.4).
The aim of this note is to refine (1.6) as follows.
then the solution (ρ,u,d) can be extended beyond T > 0.
Here BMO denotes the space of functions of bounded mean oscillations.
2 Proof of Theorem 1.2
denotes the material derivative of f.
for any ϵ > 0.
for any 0 < ϵ < 1.
whence (2.1) holds true.
This completes the proof.
Both X. Chen and J. Fan are professors. J. Fan has published more than 90 scientific papers on nonlinear partial differential equations.
The authors would like to thank the referees for careful reading and helpful comments. This article is supported by NSFC (No. 11171154).
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