Asymptotic stability of solutions to a hyperbolic-elliptic coupled system of the radiating gas on the half line

TL;DR

This paper proves the asymptotic stability of solutions to a hyperbolic-elliptic coupled system modeling radiating gas on the half line, including rarefaction waves, stationary solutions, and their superpositions, using a singular phase plane analysis and energy methods.

Contribution

It introduces a singular phase plane analysis to establish the existence and behavior of stationary solutions, especially in degenerate cases, and extends stability results to a coupled hyperbolic-elliptic system.

Findings

Proves stability of rarefaction waves, stationary solutions, and their superpositions.

Develops a singular phase plane analysis for degenerate cases.

Uses $L^2$-energy method for stability proof.

Abstract

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of the radiating gas, where the data on the boundary and at the far field state are defined as $u_-$ and $u_+$ satisfying $u_-<u_+$. For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara (SIAM J. Math. Anal. 29 (1998) 293-308) that the solution tends toward rarefaction wave or stationary solution or superposition of these two kind of waves depending on the distribution of $u_\pm$. Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of the radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: $u_-<u_+=0$ such that the system has inevitable singularities. The stability of rarefaction wave, stationary solution, and their superposition, is proved by applying the standard $L^2$-energy method.