On the Regularity Criteria for 3-D Liquid Crystal Flows in Terms of the Horizontal Derivative Components of the Pressure

DOI：10.3770/j.issn:2095-2651.2020.02.005

 作者 单位 赵玲玲 大连理工大学数学科学学院, 辽宁 大连 116024 李风泉 大连理工大学数学科学学院, 辽宁 大连 116024

本文主要研究了三维液晶流涉及到压力水平梯度量及液晶分子方向场梯度的相关正则性准则.更准确地说,方程的强解$(u,d)$可以延拓过$T$,如果下列条件成立:$$\nabla_hP\in L^{s}(0,T;L^q(\mathbb{R}^{3})),~\frac{2}{s}+\frac{3}{q}\leq\frac{5}{2},~\frac{18}{13}\leq{q}\leq 6$$和$$\nabla d\in{L^{\beta}(0,T;L^{\gamma}(\mathbb{R}^{3})),~\frac{2}{\gamma}+\frac{3}{\beta}\leq\frac{3}{4},~\frac{36}{7}\leq{\beta}\leq 12 }.$$

This paper is devoted to investigating regularity criteria for the 3-D nematic liquid crystal flows in terms of horizontal derivative components of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution $(u,d)$ can be extended beyond $T$, provided that the horizontal derivative components of the pressure $\nabla_h P=(\partial_{x_{1}} P,\partial_{x_{2}} P)$ and gradient of the orientation field satisfy $$\nabla_hP\in L^{s}(0,T;L^q(\mathbb{R}^{3})),~\frac{2}{s}+\frac{3}{q}\leq\frac{5}{2},~\frac{18}{13}\leq{q}\leq 6$$ and $$\nabla d\in{L^{\beta}(0,T;L^{\gamma}(\mathbb{R}^{3})),~\frac{2}{\gamma}+\frac{3}{\beta}\leq\frac{3}{4},~\frac{36}{7}\leq{\beta}\leq 12 }.$$