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) ∇huh∈L2p2p−3(0,T;Lp(ℝ3)),∂3d∈L2qq−3(0,T;Lq(ℝ3)),32<p≤∞,3<q≤∞$$ {\nabla}_h{u}_h\in {L}^{\frac{2p}{2p-3}}\left(0,T;{L}^p\left({\mathbb{R}}^3\right)\right),{\partial}_3d\in {L}^{\frac{2q}{q-3}}\left(0,T;{L}^q\left({\mathbb{R}}^3\right)\right),\frac{3}{2}<p\le \infty, 3<q\le \infty $$ and (ii) ∇huh∈Lq(0,T;Lp(ℝ3)),3p+2q≤1,3<p<4$$ {\nabla}_h{u}_h\in {L}^q\left(0,T;{L}^p\left({\mathbb{R}}^3\right)\right),\frac{3}{p}+\frac{2}{q}\le 1,3<p<4 $$. [ABSTRACT FROM AUTHOR]