A Refined Regularity Criterion for $3$D Liquid Crystal Equations Involving Horizontal Velocity

Abstract:

      This note investigates the global regularity of $3$D liquid crystal equations in terms of the vertical derivative of $u_h$. More precisely, we prove that if the vertical derivative of the horizontal velocity component $u_h$ satisfies $\pa_3u_h\in L^p(0,T; \R^3)$ with $\frac{2}{p}+\frac{3}{q}\le \frac{3}{2}$, $2\le p\le \infty$, then the local strong solution $(u,d)$ can be smoothly extended beyond $t=T$.