Uniqueness of Transonic Shock Solutions in a Duct for Steady Potential Flow

TL;DR

This paper proves the uniqueness of transonic shock solutions in a duct for steady potential flow, showing that for a given upstream flow, there is a unique exit pressure allowing a transonic shock, with solutions being unique up to translation.

Contribution

It establishes a new uniqueness theorem for transonic shock solutions in ducts, including cases not close to background solutions, using maximum principles and comparison methods.

Findings

Unique transonic shock solution exists for a specific exit pressure.

No solutions exist for other uniform exit pressures.

Solutions are unique modulo translation.

Abstract

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.