Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows

In this paper, we study the regularity criterion for the local smooth solution of the 3D nematic liquid crystal flows. More precisely, it is proved the smooth solution u,d can be extended beyond T provided that integral 0Tdel huhBinfinity,infinity 0+del dBinfinity,infinity 02/1+log1+del uBinfinity,infinity 0+del dBinfinity,infinity 0dt< or integral 0Tdel huhBinfinity,infinity -r4/3-2r+del dBinfinity,infinity 02/1+log1+del uBinfinity,infinity 0+del dBinfinity ,infinity 0dt<,0<= r<= 1.

Automated summary

This paper investigates the regularity criterion for the local smooth solution of 3D nematic liquid crystal flows. It is proven that the smooth solution can be extended beyond a given time T, based on certain integral conditions.