Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

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• 作者：Qiao Liu ; Yemei Wei
• 关键词：Incompressible nematic liquid crystal flows ; Navier–Stokes equations ; Blow up criteria
• 刊名：Acta Applicandae Mathematicae
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：147
• 期：1
• 页码：63-80
• 全文大小：
• 刊物主题：Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9036
• 卷排序：147

文摘

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if $$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$ or $$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$ where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).KeywordsIncompressible nematic liquid crystal flowsNavier–Stokes equationsBlow up criteriaThis work is partially supported by the National Natural Science Foundation of China (11401202), the Scientific Research Fund of Hunan Provincial Education Department (14B117), and the China Postdoctoral Science Foundation (2015M570053, 2016T90063).