Slightly Rarefied Gas: Asymptotic Theory of the Boltzmann System for Small Knudsen Numbers

Abstract

When the mean free time and the mean free path of the gas molecules become smaller, the contribution of the collision term in the Boltzmann equation becomes larger, and the velocity distribution function will approach a local Maxwellian. Then, the behavior of the gas may be considered to admit a macroscopic description, because the distribution function is determined by the five macroscopic variables. In fact, the asymptotic theory of the Boltzmann system for small mean free time and small mean free path is developed for an initial-value problem (Grad [1963a]) and for time-independent boundary-value problems in arbitrary domains (see Sone [2002]). According to it, the overall behavior of the gas is described by fluid-dynamic-type equations with initial conditions or boundary conditions given by prescribed formulas. The corrections to the overall solution are required in a thin layer near the initial state or the boundary (or initial or Knudsen layer) and in a shock layer. The correction formulas are established.