Transonic Shocks in Multidimensional Divergent Nozzles

Abstract

We establish existence, uniqueness and stability of transonic shocks for a steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity (non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Fr,chet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.