*Mathematical Methods in the Applied Sciences*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

19 Aug 2022

19 Aug 2022

19 Aug 2022

13 Sep 2022

01 Jan 2023

02 Jan 2023