20211126 刘桥 Regularity criteria for weak solutions to the 3d co-rotational Beris-Edwards system via the pressure

报告题目: Regularity criteria for weak solutions to the 3d co-rotational Beris-Edwards system via the pressure

报告人：刘桥 副教授(中南大学)

时间：2021年11月26日 (周五), 10:40-11:40 (上午)

地点：425

邀请人：周建丰

报告摘要: In this talk, we investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier--Stokes equations for the fluid velocity $\bu$ with an evolution-diffusion equations for the $Q$-tenser. Our results yield that for any positive constant $\gamma >0$, if either the negative part of the associated pressure $\Pi$ satisfies

\begin{align*}

\Pi_{-} [\ln(1+\Pi_{-})]^{1+\gamma} \in L^{\infty}(\mathbb{R}_+; L^{\frac{3}{2},\infty}(\mathbb{R}^3)),

\end{align*}

or the quantity

$ 2\Pi +|\bu|^{2}+|\nabla Q|^{2}$ satisfies

\begin{align*}

(2\Pi_{+}+|\bu|^2+|\nabla Q|^2) [\ln(1+2\Pi_{+}+|\bu|^2+|\nabla Q|^2)]^{1+\gamma} \in L^{\infty}(\mathbb{R}_+; L^{\frac{3}{2},\infty}(\mathbb{R}^3)),

\end{align*}

then the weak solution $(\bu,Q)$, to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript ``$-$" and ``$+$" denote the negative and the nonnegative part, respectively. $L^{\frac{3}{2},\infty}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space. If $Q\equiv \mathbf{0}$, then our results extend some previous known results from the theory of the 3d Navier--Stokes equations.

报告人简介：刘桥，现为中南大学数学与统计学院副教授，于2012年获中山大学博士学位；2014年-2016年在北京应用物理与计算数学研究所从事博士后研究工作；2017年8月-2018年8月访问普渡大学（Purdue University）数学系。现研究领域为不可压流体中偏微分方程如Navier--Stokes 方程组和向列型液晶流体中相关方程等的数学研究。已发表SCI论文60多篇。现主持国家自然科学基金面上项目1项。