Multidimensional Transonic Shock Waves and Free Boundary Problems

TL;DR

This paper surveys recent advances in analyzing multidimensional transonic shock waves and free boundary problems for the Euler equations, highlighting new methods for longstanding challenges in nonlinear PDEs of mixed type.

Contribution

It introduces an effective nonlinear method for solving free boundary problems related to transonic shocks in multidimensional Euler equations.

Findings

Formulation of transonic shock problems as free boundary problems.

Development of a nonlinear method to analyze these free boundary problems.

Potential applications to other nonlinear PDE free boundary problems.

Abstract

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic-hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection-diffraction, and the Prandtl-Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.