A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field

In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $(u,d)$ is regular provided that velocity component $u_3$, vorticity component ${\omega }_3$ and the horizontal derivative components of the orientation field ${\mathrm{\nabla }}_{h}d$ satisfy

$$\begin{array}{r}{\int }_0^T||u_3|{|}_{L^p}^{\frac{2p}{p-3}}+||{\omega }_3|{|}_{L^q}^{\frac{2q}{2q-3}}+||{\mathrm{\nabla }}_{h}d|{|}_{L^a}^{\frac{2a}{a-3}}\mathrm{d}\mathrm{t}<\mathrm{\infty },\\ with3<p\le \mathrm{\infty },\frac{3}{2}<q\le \mathrm{\infty },3<a\le \mathrm{\infty }.\end{array}$$