Stability of a supersonic flow about a wedge with weak shock wave

It is known that the problem of finding the streamlines of a stationary supersonic flow of a nonviscous nonheat-conducting gas in thermodynamical equilibrium past an infinite plane wedge (with a sufficiently small angle at the vertex) in theory has two solutions: a strong shock wave solution (the velocity behind the front of the shock wave is subsonic) and a weak shock wave solution (the velocity behind the front of the shock wave is generally speaking supersonic). In the present paper it is shown for a linear approximation to this problem that the weak shock wave solution is asymptotically stable in the sense of Lyapunov. Moreover, it is shown that for initial data with compact support the solution of the mixed linear problem converges in finite time to the zero solution. In the case of linear approximation 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 asymptotically stable in the sense of Lyapunov.
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