Stability condition for strong shocks in flows over an infinite planar wedge satisfying the Lopatinski condition

Abstract

We study the classical problem for a flow of uniform inviscid non-heat-conducting gas in thermodynamical equilibrium moving onto a planar infinite wedge. As it is known, theoretically this problem has solutions of two types. Solutions of the first type correspond to a strong shock when the gas velocity behind the shock front is less than the sound speed whereas solutions of the second type correspond to the case of a weak shock when the gas velocity behind the shock front is greater than the sound speed. The shock fronts are attached to the wedge vertex. We consider the linear problem for a strong shock wave provided that the Lopatinski condition holds (in a weak sense) on the shock front and the initial data are compactly, i.e. supports of the initial data are separated from the coordinate axes. Under some additional conditions on the initial data we find a solution of the generalized problem. In contrast to the previously studied case when the uniform Lopatinski condition holds on the shock front, this solution contains plane waves. The stability of the found solution is thus justified on the linear level what gives a principle possibility to realize the flow regime containing a strong shock as time increases.