The regularity for the 3-D nematic liquid crystal equations is considered in this paper, it is proved that the Leray–Hopf weak solutions (u, d) is in fact smooth, if the velocity field \(u\in L^\infty (0,T;L^{3,\infty }_x(\mathbb {R}^3))\) satisfies some addition local small condition

$$\begin{aligned} r^{-3}\left| \left\{ x\in B_r(x_0): |u(x,t_0)|>\varepsilon r^{-1}\right\} \right| \le \varepsilon , \end{aligned}$$

which is inspired by the papers [2, 35].

中文翻译：

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关于临界洛伦兹空间中液晶方程的正则性的评注

本文考虑了 3-D 向列液晶方程的规律性，证明了 Leray-Hopf 弱解 ( u ,  d ) 实际上是光滑的，如果速度场\(u\in L^\infty (0,T;L^{3,\infty }_x(\mathbb {R}^3))\)满足一些加法局部小条件

$$\begin{对齐} r^{-3}\left| \left\{ x\in B_r(x_0): |u(x,t_0)|>\varepsilon r^{-1}\right\} \right| \le \varepsilon , \end{aligned}$$

这是受到论文 [2, 35] 的启发。