报告题目：Optimal large time behavior of the compressible Bipolar Navier--Stokes--Poisson system

报告人：张映辉

报告时间：2021年7月26日  9:00-11:00

报告地点：腾讯会议，807305835

报告摘要：This topic is concerned with the Cauchy problem of the 3D compressible bipolar Navier--Stokes--Poisson (BNSP) system, and our main purpose is three--fold: First, under the assumption that $H^l\cap L^1$($l\geq 3$)--norm of the initial data is small, we prove the optimal time decay rates of the solution as well as its all--order spatial derivatives from one--order to the highest--order, which are the same as those of the compressible Navier--Stokes equations and the heat equation. Second, for well--chosen initial data, we also show the lower bounds on the decay rates. Therefore, our time decay rates are optimal. Third, we give the explicit influences of the electric field on the qualitative behaviors of solutions, which are totally new as compared to the results for the compressible Navier--Stokes (NS) system. This phenomenon is the most important difference from the compressible Navier--Stokes equations. More precisely, we show that the densities of the BNSP system converge to their corresponding equilibriums at the same $L^2$--rate $(1+t)^{-\frac{3}{4}}$ as the compressible Navier--Stokes equations, but the momentums of the BNSP system and the difference between two densities decay at the$L^2$--rate $(1+t)^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})}$ and $(1+t)^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}$ with $1\leq p\leq \frac{3}{2}$, respectively, which depend directly on the initial low frequency assumption of electric field, namely, the smallness of $\|\nabla\phi_0\|_{L^p}$.

报告人简介：张映辉，博士(后)，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，美国佐治亚理工学院和加拿大不列颠哥伦比亚大学访问学者，美国《数学评论》评论员，国际期刊《SCIREA Journal of Mathematics》编委，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2项，国家自科基金子项目2项，广西杰出青年科学基金、博士后基金等省部级项目20余项；以独立作者、第一作者或通讯作者身份在SIAM J. Math. Anal.、J. London Math. Soc.、Indiana U. Math. J.、J. Differ. Equations、P. Roy. Soc. Edinb A、Sci. China Math. 等国际著名期刊上发表SCI论文40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。