Existence and stability of perturbed transonic shocks for compressible steady potential flows

Abstract

We establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem. We establish gradient estimates and use the fixed point theorem of Caristi’s, which does not require a compactness of the map, to establish the existence result. We provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region.