三维液晶关于压力水平梯度的相关正则性准则

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■ and gradient of the orientation field satisfy■ and■.

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

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 $$\nabla d\in{L^{\beta}(0,T;L^{\gamma}(\mathbb{R}^{3})),~\frac{2}{\gamma}+\frac{3}{\beta}\leq\frac{3}{4},~\frac{36}{7}\leq{\beta}\leq 12 }.$$