Stationary Solutions of the Boltzmann Equation

Abstract

Publisher Summary This chapter discusses the stationary solutions of the Boltzmann equation. There are many literatures on the existence and stability of stationary solutions for the incompressible case but few for the compressible case that gives a better description of gas flows. The chapter presents the works in which the compressible Euler equation is solved for small c on the existence of two-dimensional isentropic, irrotational stationary flows whose stability, however, is still open. The chapter describes the exterior problem for the Boltzmann equation describing a gas flow, and shows n ≥ 3 that if c is small, then stationary solutions exist and are asymptotically stable in time. The special case c = 0, has been solved, in the large in time, for which stationary solutions are trivially given by Maxwellians. When c ≠ 0, nontrivial stationary solutions appear.