Stationary waves to viscous heat-conductive gases in half space: existence, stability and convergence rate

TL;DR

This paper studies the long-term behavior of viscous heat-conductive gases in a half space, proving the existence and stability of stationary solutions and estimating their convergence rates using energy methods.

Contribution

It establishes the existence, stability, and convergence rate of stationary solutions for viscous heat-conductive gases in a half space, employing center manifold and energy methods.

Findings

Existence of stationary solutions under small boundary data

Asymptotic stability of solutions with small initial perturbations

Convergence rate matching the decay rate of initial perturbations

Abstract

The present paper is concerned with large-time behavior of solutions to an outflow problem for an ideal polytropic model of compressible viscous gases in one-dimensional half space, and with a convergence rate of solutions toward a corresponding stationary solution. With the aid of center manifold theory, we prove the existence of the stationary solution, under a smallness condition on the boundary data. We also investigate, by employing an energy method, the time asymptotic stability of the stationary solution under suitable smallness assumptions, and estimate the convergence rate which coincides with the spatial decay rate of the initial perturbation. The proof is mainly based on a priori estimates, which are derived by a time and space weighted energy method.