Stability of a supersonic flow over a wedge containing a weak shock wave satisfying the Lopatinski condition

Abstract

We study the stability problem for a stationary supersonic flow of inviscid non-heat-conducting gas in thermodynamical equilibrium moving onto a planar infinite wedge. As it is known, this problem has two solutions: a solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and a solution with a weak shock wave (when the velocity behind the front of the shock wave is supersonic). We consider the case of a weak shock wave and we prove that if the Lopatinski condition for the shock wave holds (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed. We thus find a classical solution to this problem. Unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. For compactly supported initial data we show that the solution of the linearized problem converges to the zero solution as time tends to infinity. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense, these results complete the proof of the well-known Courant–Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable.