Two problems on the Navier-Stokes equations and the Boltzmann equation

Abstract

We study two problems on the Navier-Stokes equations and the Boltzmann equation. Chapter 2 is devoted to the study of the compressible Navier-Stokes equations for isentropic flow when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. This assumption comes from physical consideration and it also gives the well-posedness of the Cauchy problem. A global existence result is established by some new a priori estimates so that the interval for the power of the density in the viscosity coefficient is enlarged to (0,1/3). This improves previous result in this direction. The uniqueness of the solution to the problem is also given in this chapter. The Euler equations with frictional force have been extensively studied. Since there is a close relation between the Boltzmann equation and the system of fluid dynamics, in chapter 3, we study the Boltzmann equation with frictional force which is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro-micro decomposition for the Boltzmann equation introduced by Liu, Yang, and Yu through energy estimates.

Date of Award	15 Jul 2005
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Tong YANG (Supervisor)