On the global wellposedness of the inhomogeneous Navier-Stokes equations with bounded density
数学专题报告
报告题目(Title)：On the global wellposedness of the inhomogeneous Navier-Stokes equations with bounded density
报告人(Speaker)：郝田田（北京大学）
地点(Place)： 后主楼1124
时间(Time)：12月17日16:00-17:00
邀请人(Inviter)：刘彦麟
报告摘要
In this talk, I will present a comprehensive overview of our recent works on the global well-posedness of inhomogeneous Navier-Stokes equations (INS) with bounded density. Firstly, we solve Lions' density patch problem about the preseving of boundary regularity of a density patch and Lions' open problem on the 2-D uniqueness of weak solutions. Moreover, we extend Leray's 2-D global well-posedness result of weak solutions in $L^2(\R^2)$ on the classical Navier-Stokes equations (NS) to (INS) (in which case our uniqueness requires the positive lower bound of the density), and we also extend Fujita-Kato's celebrated result on the global well-posednesss of (NS) in $\dot H^{1/2}(\mathbb R^3)$ to (INS). Later on, we further improve the condition from $\dot{H}^{1/2}$ to $\dot B^{1/2}_{2,\infty}$. This gives the first existence result of the forward self-similar solution for (INS). Finally, we extend Cannone-Meyer-Planchon solutions of (NS) in $\dot{B}^{-1+\f3p}_{p,\infty}(\R^3)$ to (INS), and we prove the weak-strong uniqueness between Cannone-Meyer-Planchon solution and Lions weak solution of (INS). This is joint work with Feng Shao, Dongyi Wei, Ping Zhang and Zhifei Zhang.
