Stability of Transonic Characteristic Discontinuities in Two-Dimensional Steady Compressible Euler Flows

TL;DR

This paper proves the structural stability of transonic characteristic discontinuities in two-dimensional steady supersonic Euler flows past a convex cornered wall, under small upstream flow perturbations, using free boundary analysis and wave interaction techniques.

Contribution

It establishes the existence and stability of weak entropy solutions with Lipschitz free boundaries for transonic flows with characteristic discontinuities.

Findings

Proved stability of transonic characteristic discontinuities under small perturbations.

Constructed weak entropy solutions with Lipschitz free boundaries.

Analyzed nonlinear wave interactions using front tracking method.

Abstract

For a two-dimensional steady supersonic Euler flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic flow from the gas at rest (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic flow in $BV$. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e. characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.