Nonlinear Partial Differential Equations

Date: October 17, 2017

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Xin Liu, Texas A&M University

Title: Nonlinear PDEs Seminar

Abstract:

Title: Some gas-vacuum interface problems of compressible Navier-Stokes equations in spherically symmetric motions

Abstract:
I will talk about the well-posedness of two problems concerning the evolution of a flow connected with vacuum. The flow, or gas, connects the vacuum area in a way that the sound speed across the gas-vacuum interface has only Holder continuity. A typical example is the Lane-Emden solution for gaseous stars, where the sound speed is only 1/2-Holder continuity on the gas-vacuum interface. As pointed out by T.P. Liu in 1996, the classical hyperbolic method fails due to such singularity. Only recently, Jang and Masmoudi, Coutand, Lindblad and Shkoller independently developed some weighted energy estimates to show the well-posedness of the inviscid isentropic flows. This work is to investigate how the viscosity will help resolve such singularity. In particular, the equilibrium and the well-posedness of a model based on the thermodynamic model listed in Chandrasekhar’s book (An introduction to the study of stellar structure) is studied. Also, we investigate the global well-posedness of the Navier-Stokes equations, which allows the density and velocity to be large, the gas to connect to vacuum in a general manner but the energy to be small. This is based on my Ph.D. thesis as a student of Prof. Zhouping Xin in the Chinese University of Hong Kong, Hong Kong. ===============