Asymptotic stability of planar rarefaction wave to a 2D hyperbolic-elliptic coupled system of the radiating gas on half space

TL;DR

This paper proves that solutions to a 2D hyperbolic-elliptic system modeling radiating gas on a half space asymptotically approach a planar rarefaction wave, using energy estimates and maximum principle techniques.

Contribution

It establishes the asymptotic stability of planar rarefaction waves for a 2D hyperbolic-elliptic coupled system under small initial perturbations.

Findings

Solution converges to planar rarefaction wave as time goes to infinity.

Stability proven under smallness assumptions on initial data.

Uses div-curl decomposition, energy methods, and maximum principle.

Abstract

This paper studies the asymptotic stability of solution to an initial-boundary value problem for a hyperbolic-elliptic coupled system on two-dimensional half space, where the data on the boundary and at the far field are prescribed as $u_-$ and $u_+$. We show that the solution to the problem converges to the corresponding planar rarefaction wave for $0\le u_-<u_+$ as time tends to infinity under smallness assumptions on the initial perturbation and wave strength. These results are based on the analysis of div-curl decomposition, the standard $L^2$-energy method, $L^1$-estimate, and the monotonicity of profile is given by the maximum principle.