Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier–Stokes equations

Abstract

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t = 0 ) as ϵ → 0 . A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier–Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier–Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.