Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow

TL;DR

This paper proves the uniform convexity of self-similar transonic shocks in potential flow, providing a new geometric framework applicable to classical shock problems like reflection-diffraction and Prandtl-Meyer reflection.

Contribution

It develops a general framework to establish the convexity of transonic shocks as free boundaries in self-similar potential flow, advancing the mathematical understanding of shock stability and structure.

Findings

Proved uniform convexity of self-similar transonic shocks.

Applied the framework to classical shock problems.

Explored nonlocal properties of solutions and free boundaries.

Abstract

We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid motions in continuum mechanics but also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the Euler equations for self-similar potential flow separates elliptic (subsonic) and hyperbolic (supersonic) phases of the self-similar solution of the corresponding nonlinear partial differential equation in a domain under consideration, in which the location of the transonic shock is apriori unknown. We first develop a general framework under which self-similar transonic shocks, as free boundaries, are proved to be uniformly convex, and then apply this framework to prove the uniform convexity of transonic shocks in the two longstanding fundamental shock problems -- the shock reflection-diffraction by wedges and the Prandtl-Meyer reflection for supersonic flows past solid ramps. To achieve this, our approach is to exploit underlying nonlocal properties of the solution and the free boundary for the potential flow equation.