  We consider the problem of 2D supersonic flow onto a solid wedge, or
equivalently in a concave corner formed by two solid walls. For mild corners,
there are two possible steady state solutions, one with a strong and one with a
weak shock emanating from the corner. The weak shock is observed in supersonic
flights. A long-standing natural conjecture is that the strong shock is
unstable in some sense.
  We resolve this issue by showing that a sharp wedge will eventually produce
weak shocks at the tip when accelerated to a supersonic speed. More precisely
we prove that for upstream state as initial data in the entire domain, the
time-dependent solution is self-similar, with a weak shock at the tip of the
wedge. We construct analytic solutions for self-similar potential flow, both
isothermal and isentropic with arbitrary $\gamma\geq 1$.
  In the process of constructing the self-similar solution, we develop a large
number of theoretical tools for these elliptic regions. These tools allow us to
establish large-data results rather than a small perturbation. We show that the
wave pattern persists as long as the weak shock is supersonic-supersonic; when
this is no longer true, numerics show a physical change of behaviour. In
addition we obtain rather detailed information about the elliptic region,
including analyticity as well as bounds for velocity components and shock
tangents.
