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

TL;DR

This paper proves the uniqueness and stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows, extending previous existence results to include these properties under small perturbations.

Contribution

It establishes the uniqueness and $L^1$-stability of transonic characteristic discontinuities, a significant advancement in understanding free boundary problems in Euler flows.

Findings

Proved the solution's uniqueness for transonic characteristic discontinuities.

Established $L^1$-stability under small perturbations.

Extended previous existence results to include stability and uniqueness.

Abstract

In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right-wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic-hyperbolic composite-mixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and $L^1$--stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates.