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

Abstract:

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