Uniqueness of transonic shock solutions in general approximate nozzles for steady potential flow

TL;DR

This paper proves the uniqueness of transonic shock solutions in approximate nozzles modeled as a 2D Riemannian manifold, extending results from the sphere to more general geometries using maximum principles.

Contribution

It establishes the first uniqueness theorem for transonic shocks in general approximate nozzles modeled by elliptic-hyperbolic PDEs, extending prior results on the sphere.

Findings

Unique transonic shock solutions exist for given upstream flow and exit pressure.

The solutions are unique modulo a translation in the sphere case.

The method applies maximum principles to a free boundary problem in mixed-type PDEs.

Abstract

We study the uniqueness of solutions with a transonic shock in a two-dimensional Riemannian manifold with a special metric, which can be regarded as an approximate model of the general physical nozzles, within a class of transonic shock solutions for steady potential flow. We first prove the uniqueness of these solutions on the unit 2-sphere: for given uniform supersonic upstream flow at the entry, there exists a unique uniform pressure at the exit such that a transonic shock solution exists in the sphere, which is unique modulo a translation. A similar result is then extended to a class of manifolds. Mathematically, it is equivalent to showing a uniqueness theorem for a free boundary problem of a second-order elliptic-hyperbolic mixed-type partial differential equation in the general approximate nozzles. The proof is based on the maximum/comparison principle with a suitable special transonic shock solution as a comparison function.