$L^\infty$-Stability of Continuous Shock Waves in a Radiating Gas Model

Abstract

In the present article, we study the asymptotic stability in $L^\infty$-topology of shock waves in a model system of radiating gases. It is known that the system admits discontinuous shock waves if the shock strength is strictly above a threshold value of $\sqrt{2}$, while if it is below (subcritical case) or equal to (critical case) $\sqrt{2}$, shock waves are continuous [Kawashima and Nishibata, SIAM J. Math. Anal., 30 (1998), 95--117]. We prove that all subcritical shock waves are stable to piecewise smooth perturbations of small amplitude. The stability of subcritical shock waves is robust in the sense that it is not affected by possible collisions of discontinuities contained in initial data and the solutions converge to shock waves beyond such events. Sufficient conditions for occurrence and nonoccurrence of collision of discontinuities are both given as well. In the meantime, the critical shock wave blows up the first order derivative if certain types of perturbations are added however small the perturbations may be. Some conditional stability results are also addressed which are applicable for both subcritical and critical cases. The results imply an optimality of a blowup criterion given by Kawashima and Nishibata [Math. Models. Methods Appl. Sci., 9 (1999), pp. 69--91].