A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field

Abstract

<abstract><p>In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t&lt;\infty,\nonumber \\ with\ \ 3&lt; p\leq\infty,\ \frac{3}{2}&lt; q\leq\infty,\ 3&lt; a\leq\infty. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> </abstract>