This paper is devoted to investigating regularity criteria for the nematic liquid crystal system in terms of one derivative component 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 one derivative component of the pressure ${\mathrm{\partial }}_{x_3}P$ and gradient of the orientation field satisfy $\begin{array}{r}{\mathrm{\partial }}_{x_3}P\in L^s(0,T;L^q({\mathbb{R}}^3\left)\right),\frac{2}{s}+\frac{3}{q}<\frac{29}{10},\frac{30}{23}<q\le \frac{10}{3},\\ {\mathrm{\partial }}_{x_3}d\in L^{\gamma }(0,T;L^{\text{β}}({\mathbb{R}}^3\left)\right),\frac{2}{\gamma }+\frac{3}{\text{β}}<\frac{19}{20},\frac{60}{13}<\text{β}\le \frac{20}{3}\end{array}$ and ${\mathrm{\nabla }}_{h}d\in L^{{\gamma }_1}(0,T;L^{{\text{β}}_1}({\mathbb{R}}^3\left)\right),\frac{2}{{\gamma }_1}+\frac{3}{{\text{β}}_1}\le \frac{3}{4}+\frac{1}{2{\text{β}}_1},\frac{60}{13}<{\text{β}}_1\le \frac{20}{3}.$

中文翻译：

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涉及压力的一种微分分量的液晶系统的规律性判据

本文致力于研究向列液晶系统的规则性标准，该标准是根据取向场的压力和梯度的一个导数分量。更准确地说，我们主要证明了强解$(你,d)$可以扩展到T之外，前提是压力的一个导数分量${\mathrm{\partial }}_{X_3}磷$和方向场的梯度满足$\begin{array}{r}{\mathrm{\partial }}_{X_3}磷\in {\mathrm{大号}}^s(0,吨;{\mathrm{大号}}^q({\mathbb{R}}^3\left)\right),\frac{2}{s}+\frac{3}{q}<\frac{29}{10},\frac{30}{23}<q\le \frac{10}{3},\\ {\mathrm{\partial }}_{X_3}d\in {\mathrm{大号}}^{\gamma }(0,吨;{\mathrm{大号}}^{\text{β}}({\mathbb{R}}^3\left)\right),\frac{2}{\gamma }+\frac{3}{\text{β}}<\frac{19}{20},\frac{60}{13}<\text{β}\le \frac{20}{3}\end{array}$和${\mathrm{\nabla }}_{H}d\in {\mathrm{大号}}^{{\gamma }_1}(0,吨;{\mathrm{大号}}^{{\text{β}}_1}({\mathbb{R}}^3\left)\right),\frac{2}{{\gamma }_1}+\frac{3}{{\text{β}}_1}\le \frac{3}{4}+\frac{1}{2{\text{β}}_1},\frac{60}{13}<{\text{β}}_1\le \frac{20}{3}.$