Abstract

In this talk, we will review the past developments on the solutions of the compressible Navier-Stokes equations and reveal the three hidden structures which linked the weak solution to the strong one. Based on these observations, we proved the Nash's conjecture in 1958s and establish global exsitence theory for both isentropic and heat-conductive compressible Navier-Stokes equations. Moreover, for the 3D compressible Navier-Stokes equations, we will show the existence of local weak solutions with higher regularity and local strong solutions with lower regularity. Also, we will mention the recent results on the blowup of the local strong solutions to the MHD equations in finite time and global existence of weak solutions of the compressible Navier-Stokes equations in bounded domains under Dirichlet boundary conditions.

Speaker Intro

黄祥娣，中国科学院数学与系统科学研究院研究员，主要研究高维可压缩Navier-Stokes方程。他的成果揭示了等熵和非等熵可压缩流体允许真空初值的光滑解在有限时刻爆破的机制，彻底解决并推广了诺贝尔奖得主J.Nash在1958年提出的关于可压缩流体光滑解爆破机制的猜想。在此基础上，他首次对等熵和非等熵流体建立了允许真空初值的整体光滑解和弱解，因此部分解决了菲尔兹奖得主P.L.Lions关于理想多方气体整体弱解存在性的猜想，其成果荣获2017年首届和2020年第四届世界华人数学家联盟年会的五年最佳论文奖，并在2022年第九届世界华人数学家大会做一小时报告，同年主持了第二批中科院稳定支持基础研究青年团队计划。