Two regularity criteria of solutions to the liquid crystal flows

In this paper, we derive two regularity criteria of solutions to the nematic liquid crystal flows. More precisely, we prove the local smooth solution $(u, d)$ is regular if and only if one of the following two conditions is satisfied: (i) $\nabla_{h} u_{h}\in L^{\frac{2p}{2p-3}}(0,T; L^{p}(\mathbb{R}^{3})),\ \partial_{3} d\in L^{\frac{2q}{q-3}}(0,T; L^{q}(\mathbb{R}^{3})),\ \frac{3}{2}< p\leq\infty,\ 3< q\leq\infty$; and (ii) $\nabla_{h} u_{h}\in L^{q}(0,T; L^{p}(\mathbb{R}^{3})),\ \frac{3}{p}+\frac{2}{q}\leq 1, \ 3