  When a plane shock hits a two-dimensional wedge head on, it experiences a
reflection-diffraction process, and then a self-similar reflected shock moves
outward as the original shock moves forward in time. The experimental,
computational, and asymptotic analysis has indicated that various patterns
occur, including regular reflection and Mach reflection. The von Neumann
conjectures on the transition from regular to Mach reflection involve the
existence, uniqueness, and stability of regular shock reflection-diffraction
configurations, generated by concave cornered wedges for compressible flow. In
this paper, we discuss some recent developments in the study of the von Neumann
conjectures. More specifically, we present our recent results of the uniqueness
and stability of regular shock reflection-diffraction configurations governed
by the potential flow equation in an appropriate class of solutions. We first
show that the transonic shocks in the global solutions obtained in Chen-Feldman
[19] are convex. Then we establish the uniqueness of global shock
reflection-diffraction configurations with convex transonic shocks for any
wedge angle larger than the detachment angle or the critical angle. Moreover,
the stability of the solutions with respect to the wedge angle is also shown.
Our approach also provides an alternative way of proving the existence of the
admissible solutions established first in [19].
