On the Regularity Criteria for 3-D Liquid Crystal Flows in Terms of the Horizontal Derivative Components of the Pressure
Received:July 08, 2019  Revised:October 29, 2019
Key Word: regularity criteria   nematic liquid crystal
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11571057).
 Author Name Affiliation Lingling ZHAO School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China Fengquan LI School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
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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 $$\nabla d\in{L^{\beta}(0,T;L^{\gamma}(\mathbb{R}^{3})),~\frac{2}{\gamma}+\frac{3}{\beta}\leq\frac{3}{4},~\frac{36}{7}\leq{\beta}\leq 12 }.$$