Structural stability of the transonic shock problem in a divergent three dimensional axisymmetric perturbed nozzle

TL;DR

This paper proves the structural stability of transonic shocks in a three-dimensional axisymmetric Euler system with swirl, addressing singularities near the shock boundary and axis, using a novel Lagrangian transformation.

Contribution

It introduces an invertible Lagrangian transformation to handle singularities and proves stability of transonic shocks under various perturbations in a 3D axisymmetric setting.

Findings

Proved stability of transonic shocks with swirl in 3D axisymmetric flows.

Developed a Lagrangian transformation to straighten streamlines.

Addressed singularities near shock boundary and axis.

Abstract

In this paper, we prove the structural stability of the transonic shocks for three dimensional axisymmetric Euler system with swirl velocity under the perturbations for the incoming supersonic flow, the nozzle boundary, and the exit pressure. Compared with the known results on the stability of transonic shocks, one of the major difficulties for the axisymmetric flows with swirls is that corner singularities near the intersection point of the shock surface and nozzle boundary and the artificial singularity near the axis appear simultaneously. One of the key points in the analysis for this paper is the introduction of an invertible Lagrangian transformation which can straighten the streamlines in the whole nozzle and help to represent the solutions of transport equations explicitly.