  When an upstream steady uniform supersonic flow impinges onto a symmetric
straight-sided wedge, governed by the Euler equations, there are two possible
steady oblique shock configurations if the wedge angle is less than the
detachment angle -- the steady weak shock with supersonic or subsonic
downstream flow (determined by the wedge angle that is less or larger than the
sonic angle) and the steady strong shock with subsonic downstream flow, both of
which satisfy the entropy condition. The fundamental issue -- whether one or
both of the steady weak and strong shocks are physically admissible solutions
-- has been vigorously debated over the past eight decades. In this paper, we
survey some recent developments on the stability analysis of the steady shock
solutions in both the steady and dynamic regimes. For the static stability, we
first show how the stability problem can be formulated as an initial-boundary
value type problem and then reformulate it into a free boundary problem when
the perturbation of both the upstream steady supersonic flow and the wedge
boundary are suitably regular and small, and we finally present some recent
results on the static stability of the steady supersonic and transonic shocks.
For the dynamic stability for potential flow, we first show how the stability
problem can be formulated as an initial-boundary value problem and then use the
self-similarity of the problem to reduce it into a boundary value problem and
further reformulate it into a free boundary problem, and we finally survey some
recent developments in solving this free boundary problem for the existence of
the Prandtl-Meyer configurations that tend to the steady weak supersonic or
transonic oblique shock solutions as time goes to infinity. Some further
developments and mathematical challenges in this direction are also discussed.
