Courant-Friedrichs’ Hypothesis and Stability of the Weak Shock Wave Satisfying the Lopatinski Condition

We are studying the problem of a stationary supersonic flow of an inviscid non-heat-conducting gas in thermodynamical equilibrium onto a planar infinite wedge. It is known that theoretically this problem has two solutions: the solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and the solution with a weak shock wave (when, generally speaking, the velocity behind the front of the shock wave is supersonic). In the present paper, the case of a weak shock wave is studied. It is proved that if the Lopatinski condition for the shock wave is satisfied (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed, and its classical solution is found. In this case, unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. It is shown that for compactly supported initial data the solution of the linearized problem converges in finite time to the zero solution. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense these results complete the verification of the well-known Courant-Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable.