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 that the local smooth solution (u,d)$$ \left(u,d\right) $$ is regular if and only if one of the following two conditions is satisfied: (i) backward difference huh is an element of L2p2p-3(0,T;Lp(Double-struck capital R3)), partial differential 3d is an element of L2qq-3(0,T;Lq(Double-struck capital R3)),32

{\frac{2p}{2p-3}}\left(0,T;{L}<^>p\left({\mathbb{R}}<^>3\right)\right),{\partial}_3d\in {L}<^>{\frac{2q}{q-3}}\left(0,T;{L}<^>q\left({\mathbb{R}}<^>3\right)\right),\frac{3}{2}q\left(0,T;{L}<^>p\left({\mathbb{R}}<^>3\right)\right),\frac{3}{p}+\frac{2}{q}\le 1,3.