Transonic Shocks In Multidimensional Divergent Nozzles

TL;DR

This paper proves the existence, uniqueness, and stability of transonic shocks in multidimensional divergent nozzles for steady compressible flow, using advanced PDE techniques and addressing weak regularity conditions.

Contribution

It introduces novel methods for solving free boundary problems involving elliptic and transport equations with weak regularity and discontinuous boundary conditions.

Findings

Established existence and uniqueness of transonic shocks in multidimensional nozzles.

Developed a weak implicit mapping theorem for PDEs with non-Lipschitz velocity fields.

Provided regularity results for elliptic PDEs with discontinuous oblique boundary conditions.

Abstract

We establish existence, uniqueness and stability of transonic shocks for 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 Frechet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.