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

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

“…The coefficient 1 0 D  and, moreover, 1 (for example, this is true for the polytropic gas [10]; some examples of state equations satisfying (8) are considered in [15]). Assume that the solution of problems (1)-(5) is not just continuous but also has second derivatives which are continuous up to the boundary of the domain.…”

“…The Courant-Friedrichs' hypothesis was verified in [4][5][6][7] (but this conclusion was based only on some qualitative reasons). A strict mathematical justification (and this is very important) of this statement for the linearized problem appeared in recent years in [8][9][10][11][12]. Briefly speaking, it was shown in [8][9][10][11][12] that in the case of a strong shock wave (for compactly supported initial data!)…”

“…A strict mathematical justification (and this is very important) of this statement for the linearized problem appeared in recent years in [8][9][10][11][12]. Briefly speaking, it was shown in [8][9][10][11][12] that in the case of a strong shock wave (for compactly supported initial data!) the perturbation arrives to the wedge's vertex as time increases having the growth r  ( 0   ) or a logarithmic growth in space variables, and this causes instability of the steady-state solution under consideration.…”

“…It is assumed that in the both cases the shock satisfies the well-known uniform Lopatinski condition [13]. The present work is a continuation of [10]. We consider the more general case when the Lopatinski condition on the shock wave is satisfied only in a weak sense [13], i.e.…”

“…In particular, this makes finding the classical solution of this problem more difficult. The key point of the work is the analysis of an explicit form of the solution and we extensively use the technique developed in [10].…”