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A remark on regularity of liquid crystal equations in critical Lorentz spaces

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Abstract

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].

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  1. School of Mathematic Sciences, Fudan University, Shanghai, China

    Xiangao Liu, Yueli Liu & Zixuan Liu

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  1. Xiangao Liu
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This work was supported partly by NSFC Grant 11971113, 11631011.

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Liu, X., Liu, Y. & Liu, Z. A remark on regularity of liquid crystal equations in critical Lorentz spaces. Annali di Matematica 200, 1709–1734 (2021). https://doi.org/10.1007/s10231-020-01056-4

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