Uniqueness of Transonic Shock Solutions for Two-Dimensional Steady Compressible Euler Flows in an Expanding Nozzle

TL;DR

This paper proves the uniqueness of transonic shock solutions in expanding nozzles for steady compressible Euler flows by establishing a priori estimates and using contraction arguments under small perturbations.

Contribution

It introduces a novel approach to demonstrate the uniqueness of transonic shock solutions without requiring the shock to be a small perturbation of a uniform state.

Findings

Uniqueness of transonic shock solutions under certain boundary conditions.

A new pressure condition across the shock front enabling a priori estimates.

Verification of solution coincidence via contraction mapping.

Abstract

In this paper, we are trying to show the uniqueness of transonic shock solutions in an expanding nozzle under certain conditions and assumptions on the boundary data and the shock solution. The idea is to compare two transonic shock solutions and show that they should coincide if the perturbation of the nozzle is sufficiently small. To this end, a condition on the pressure of the flow across the shock front is proposed, such that a priori estimates for the subsonic flow behind the shock front could be established without the assumption that it is a small perturbation of the unperturbed uniform subsonic state. With the help of these estimates, the uniqueness of the position of the intersection point between the shock front and the nozzle boundary could be further established by demonstrating the monotonicity of the solvability condition for the elliptic sub-problem of the subsonic flow behind the shock front. Then, via contraction arguments, two transonic shock solutions could be verified to coincide as the perturbation is small, which leads to the uniqueness of the transonic shock solution.