Stability and Asymptotic Behavior of Transonic Flows Past Wedges for the Full Euler Equations

TL;DR

This paper proves the existence, uniqueness, stability, and asymptotic behavior of steady transonic flows with shocks past wedges governed by the full Euler equations, using free boundary and elliptic PDE techniques.

Contribution

It establishes the stability and asymptotic properties of transonic shocks for the full Euler equations, treating the shock as a free boundary and employing Schauder fixed point methods.

Findings

Existence and uniqueness of steady transonic flows with shocks.

Stability of weak and strong transonic shocks under perturbations.

Asymptotic analysis revealing characteristics of the full Euler equations.

Abstract

The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of both the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted H\"older norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.