Chapter 5 Stationary problem of Boltzmann equation

Abstract

This chapter discusses the stationary problem of the Boltzmann equation. It is well known that the Boltzmann equation has a special stationary solution called the “Maxwellian.” The Maxwellian is a universal distribution function, which appears when the gas attains an equilibrium state. If an external forcing is exerted on the gas, the non-Maxwellian steady state may persist. This external forcing may be caused through the boundary of the vessel containing the gas, the external force field, and the external gas source. The chapter describes three stationary problems: (1) the half-space problem, (2) the exterior problem, and (3) the time-periodic flow problem. It discusses the study of the half-space problem under the Dirichlet boundary condition, which arises in the analysis of the complex interaction of the gas and the solid wall of the vessel for the physical background. The main feature of this problem is that it is not an unconditionally solvable problem, and the number of the solvability conditions to be imposed on the Dirichlet data varies with the Mach number of the far field. The chapter studies the exterior problem for the flow past an obstacle and constructs the Boltzmann flow under the smallness assumption on the Mach number of the far field. The Boltzmann shock profile of transonic and supersonic flows induced by the obstacle is one of the challenging open problems in the mathematical theory of the Boltzmann equation. The chapter also discusses the inhomogeneous Boltzmann equation with a time-periodic source term. This is a simplest model problem for the generation and propagation of the sound wave in the gas.