A Refined Regularity Criterion for $3$D Liquid Crystal Equations Involving Horizontal Velocity
Received:October 18, 2020  Revised:June 26, 2021
Key Word: liquid crystal flow   regularity criterion   local strong solution
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11961032; 11971209) and the Natural Science Foundation of Jiangxi Province (Grant No.20191BAB201003).
 Author Name Affiliation Xiaoli CHEN School of Mathematics and Statistics, Jiangxi Normal University, Jiangxi 330022, P. R. China Xiujuan ZHA School of Mathematics and Statistics, Jiangxi Normal University, Jiangxi 330022, P. R. China
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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 $t=T$.