Structural Stability of Transonic Shock Flows with an External Force

TL;DR

This paper investigates the stability of transonic shock flows in flat nozzles under external forces, establishing existence, uniqueness, and stability results for these flows using advanced mathematical techniques.

Contribution

It introduces a novel analysis of how external forces stabilize transonic shocks and proves their structural stability under perturbations, using deformation-curl decomposition methods.

Findings

External force stabilizes transonic shocks in nozzles.

Existence and uniqueness of shock solutions are established.

Shock position is uniquely determined and stable under pressure perturbations.

Abstract

This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shoc solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed in \cite{WengX2019} to deal with the transonic shock problem.