Asymptotic stability of planar stationary solution to a 2D hyperbolic-elliptic coupled system of the radiating gas in half space

TL;DR

This paper proves the asymptotic stability of planar stationary solutions for a 2D hyperbolic-elliptic radiating gas system in half space, showing convergence rates under small initial perturbations using energy methods.

Contribution

It establishes the stability and convergence rates of solutions to a coupled hyperbolic-elliptic system in half space, extending understanding of radiating gas dynamics.

Findings

Solution converges to stationary state as time tends to infinity.

Convergence rate is $t^{-rac{eta}{2}-rac{1}{4}}$ for non-degenerate case.

Convergence rate is $t^{-rac{1}{4}}$ for degenerate case.

Abstract

This paper is concerned with the asymptotic stability of planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic-elliptic coupled system of the radiating gas in half space. We show that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. These results are proved by the standard $L^2$-energy method. Moreover, we prove that the solution $(u,q)$ converges to the corresponding planar stationary solution at the rate $t^{-\alpha/2-1/4}$ for non-degenerate case, and $t^{-1/4}$ for degenerate case. The proof is based on the time and space weighted energy method.