A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows

In this paper, we obtain a regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows. That is the smooth solution $(u,d)$ can be extended beyond T, provided that ${\int }_0^T(||u_h|{|}_{{\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^0}^2+||\mathrm{\nabla }d|{|}_{{\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^0}^2)\mathrm{d}\mathrm{t}<\mathrm{\infty }$.