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

Abstract

This paper studies the asymptotic stability of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupling system on the two-dimensional half-space, where the data on the boundary and at the far field are prescribed as $u_{-}$ and $u_{+}$, respectively. 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. To the best of our knowledge, the stability results of planar rarefaction waves on half-space focus primarily on the single viscous conservation law because the rarefaction wave (one-dimensional diffusion wave) of the corresponding one-dimensional problem to scalar viscous conservation law is known. In other words, for a general high-dimensional system of equations, we cannot obtain the stability of planar rarefaction waves on half-space because we cannot construct the rarefaction wave of the corresponding one-dimensional problem. In this paper, we use the structure of the hyperbolic-elliptic coupling system to obtain the monotonic rarefaction wave of the corresponding one-dimensional hyperbolic-elliptic coupling system, and hence give the stability of the planar rarefaction wave on half-space. This can be viewed as the first result for the system of equations on the stability of planar rarefaction waves on half-space.