报告题目: Global Stability and Non--Vanishing Vacuum States of 3D Compressible Navier--Stokes Equations
报告人：张映辉教授 (广西师范大学)
时间： 2023年2月22日, 08:30-9:30 (上午)
报告地点: 数学院425
邀请人：周建丰
报告摘要: We investigate global stability and non--vanishing vacuum states of large solutions to the compressible Navier—Stokes equations on the torus $\mathbb{T}^3$, and the main purpose of this work is three-fold: First, under the assumption that the density $\rho({\bf{x}}, t)$ verifies $\sup_{t\geq 0}\|\rho(t)\|_{L^\infty}\leq M$, it is shown that the solutions converge to
equilibrium state exponentially in $L^2$--norm. In contrast to previous related works where the density has uniform positive lower and upper bounds, this gives the first stability result for large strong solutions of the 3D compressible Navier--Stokes equations in the presence of vacuum. Second, by employing some new thoughts, we also show that the density converges to its equilibrium state exponentially in $L^\infty$--norm if additionally the initial density $\rho_0({\bf{x}})$ satisfies $\inf_{{\bf{x}}\in \mathbb{T}^3}\rho_0({\bf{x}})\geq c_0>0$. Finally, we prove that the vacuum state will persist for any time provided that the initial density contains vacuum, which is different from the previous work of [H. L. Li et al., Commun. Math. Phys., 281 (2008), 401--444], where the authors showed that any vacuum state must vanish within finite time for the free boundary problem of the 1D compressible Navier--Stokes equations with density—dependent viscosity $\mu(\rho)=\rho^\alpha$ with $\alpha>1/2$. This phenomenon implies the different behaviors for Navier--Stokes equations with different types of viscous effects, namely, degenerate or not.
报告人简介：张映辉，教授，博士生导师，广西杰出青年基金获得者，广西高等学校中青年骨干教师，广西师范大学A类漓江学者，广西高校数学模型及其应用重点实验室主任，广西师范大学应用数学研究所所长，现任广西师范大学数学与统计学院副院长。主要研究方向为偏微分方程理论及其应用。主持国家自然科学基金面上项目、青年项目和天元项目各1项，广西杰出青年科学基金、博士后基金等多项省部级项目；在SIAM J. Math. Anal.、 J. London Math. Soc.、Indiana U. Math. J.、 Nonlineaity、 J. Differ. Equations、Sci. China Math. 等期刊上发表学术论文20余篇；出版英文学术专著2部；获省自然科学奖和市科技进步奖各1项。
