On the regularity criteria for liquid crystal flows involving the
gradient of one velocity component
Abstract
In this paper, we show a regularity criteria for three dimensional
nematic liquid crystal flows. More precisely, we prove that the strong
solution $(u,d)$ can be extended beyond $T$, provided
$\nabla u_{3}\in{L^{p}(0,T;
L^{q}(\mathbb{R}^{3}))}$,
$\frac{2}{p}+\frac{3}{q}\leq{\frac{19}{12}+\frac{1}{2q}}(\frac{30}{19}3)$
with some conditions about the orientation field
$\nabla_{h}d\in{L^{\alpha}(0,T;
L^{\beta}(\mathbb{R}^{3}))}$,
$\frac{2}{\alpha}+\frac{3}{\beta}\leq{\frac{3}{4}+\frac{1}{2\beta}}(\beta>{\frac{10}{3}})$.