Stability of reflection and refraction of shocks on interface

Abstract

In this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media. The attack will produce a reflected wave and a refracted wave, and also let the interface deflected. In this paper we will mainly study the case, when the reflected wave is a shock, and the flow between the reflected wave and the refracted shock is relatively subsonic. Our result indicates that the wave structure and the flow field for the reflection–refraction problem in this case is conditionally stable. To describe the motion of the fluid we use the inviscid Euler system as the mathematical model. The reflection–refraction problem can be reduced to a free boundary value problem, where the unknown reflected shock and refracted shock are free boundaries, and the deflected interface is also to be determined. In the proof of the existence and the stability of the corresponding wave structure we apply the Lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic–hyperbolic composite system in its principal part. Meanwhile, some efficient weighted Sobolev estimates are established to derive the existence for corresponding nonlinear problems.