The metastability phenomen for a radiative gas model

Abstract

We study the time-asymptotic behaviour of solutions to a singular perturbation of the inviscid Burgers problem describing sound propagation in a radiating gas. This radiating gas model exhibits the interesting phenomenon known as metastability that has been first identified for the viscous Burgers equation. Precisely, on an initial time span, that scales linearly in the inverse of the perturbation parameter, we can quantify the L1-distance of the solution to the so-called sharp N-wave solution of the inviscid Burgers problem and to a diffusive N-wave. On a second long-time stage that scales exponentially in the inverse of the perturbation parameter, the solutions still remain close to a diffusive N-wave before finally collapsing into diffusion waves. Besides our analysis of the radiating gas model, we also provide an alternative description of metastability in the viscous Burgers model which, unlike other studies, relies on L1-estimates and does not require the passage to similarity variables.