Regularity Criterion for the Liquid Crystal Flow in
$\mathbb{R}^3$}

We obtain that a weak solution $(u,d)$ to the Liquid Crystal system is
strong, if any two components of $u$ and $\nabla d$
satisfy Ladyzhenskaya-Prodi-Serrin’s condition, that is
$$\|u^1\|_{L^{r,s}}+\|u^2\|_{L^{r,s}}+\|\nabla
d\|_{L^{r,s}}\leq
\infty,
\qquad\frac{3}{r}+\frac{2}{s}\leq1.$$
We also prove that the velocity $u$ is bounded locally if any two
comonents of $u$ and $\nabla d$ belong to
$L^{6,\infty}$ and $u$ belongs to
$L^{2+\delta}$
($\delta>0$).