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<\infty,\nonumber \\ with\ \ 3< p\leq\infty,\ \frac{3}{2}< q\leq\infty,\ 3< a\leq\infty. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> </abstract>

Two regularity criteria of solutions to the liquid crystal flows

“…It is obvious that system () reduces to the well‐known Navier–Stokes equations when the orientation field $$$ d\equiv 1 $$$. Therefore, in view of the results of regularity criterion for the 3D Navier–Stokes equations, some Prodi–Serrin type regularity criteria of the nematic liquid crystal flows have been obtained based on the velocity field $$$ u $$$, the gradient of the velocity field $$$ \nabla u $$$, and their components; see earlier studies 3–12 and the references therein.…”

Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows

“…Hence much eforts have been paid to study the regularity criteria to extend local solutions. For the regularity criteria readers may refer to [5][6][7][8][9][10][11][12][13] and references therein.…”