Two regularity criteria of solutions to the liquid crystal flows

Abstract

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 false(u,dfalse)$$ \left(u,d\right) $$ is regular if and only if one of the following two conditions is satisfied: (i) ∇huh∈L2p2p−3false(0,T;Lpfalse(ℝ3false)false),0.1em∂3d∈L2qq−3false(0,T;Lqfalse(ℝ3false)false),0.1em32<p≤∞,0.1em3<q≤∞$$ {\nabla}_h{u}_h\in {L}&#x0005E;{\frac{2p}{2p-3}}\left(0,T;{L}&#x0005E;p\left({\mathbb{R}}&#x0005E;3\right)\right),{\partial}_3d\in {L}&#x0005E;{\frac{2q}{q-3}}\left(0,T;{L}&#x0005E;q\left({\mathbb{R}}&#x0005E;3\right)\right),\frac{3}{2}&lt;p\le \infty, 3&lt;q\le \infty $$ and (ii) ∇huh∈Lqfalse(0,T;Lpfalse(ℝ3false)false),0.1em3p+2q≤1,0.1em3<p<4$$ {\nabla}_h{u}_h\in {L}&#x0005E;q\left(0,T;{L}&#x0005E;p\left({\mathbb{R}}&#x0005E;3\right)\right),\frac{3}{p}&#x0002B;\frac{2}{q}\le 1,3&lt;p&lt;4 $$.