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

Abstract

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 $\left(u,d\right)$ can be extended beyond $T$ provided that ${\displaystyle {\int }_0^T\left({‖{\nabla }_h{u}_h‖}_{{\dot{B}}_{\infty ,\infty }^0}+{‖\nabla d‖}_{{\dot{B}}_{\infty ,\infty }^0}^2\left)/\right.\sqrt{1+\log \left(1+{‖\nabla u‖}_{{\dot{B}}_{\infty ,\infty }^0}+{‖\nabla d‖}_{{\dot{B}}_{\infty ,\infty }^0}\right)}\text{d}t<\infty \right.}$ or ${\displaystyle {\int }_0^T\left({‖{\nabla }_h{u}_h‖}_{{\dot{B}}_{\infty ,\infty }^{-r}}^{4/3-2r}+{‖\nabla d‖}_{{\dot{B}}_{\infty ,\infty }^0}^2\left)/\right.\sqrt{1+\log \left(1+{‖\nabla u‖}_{{\dot{B}}_{\infty ,\infty }^0}+{‖\nabla d‖}_{{\dot{B}}_{\infty ,\infty }^0}\right)}\text{d}t<\infty ,0\le r\le 1\right.}$ .