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 19, 2019
Key Word: Regularity criteria   nematic liquid crystal  
Fund ProjectL:The National Natural Science Foundation of China (Grant No.11571057)
Author NameAffiliationE-mail
Lingling ZHAO Dalian University of Technology linglingz@126.com 
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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 \begin{equation*} \nabla_hP\in L^{s}(0,T;L^q(\mathbb{R}^{3})),\quad \frac{2}{s}+\frac{3}{q}\leq\frac{5}{2},\quad \frac{18}{13}\leq{q}\leq 6 \end{equation*} and \begin{equation*} \nabla d\in{L^{\beta}(0,T;L^{\gamma}(\mathbb{R}^{3})),\quad \frac{2}{\gamma}+\frac{3}{\beta}\leq\frac{3}{4},\quad \frac{36}{7}\leq{\beta}\leq 12 }. \end{equation*}
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