Free Boundary Problems in Shock Reflection/Diffraction and Related Transonic Flow Problems

TL;DR

This paper reviews how complex shock reflection and diffraction phenomena in high-speed flows can be modeled as free boundary problems, highlighting recent mathematical advances and open challenges in the field.

Contribution

It formulates key shock problems as free boundary problems and discusses recent progress and open issues in their mathematical analysis.

Findings

Formulation of shock reflection/diffraction problems as free boundary problems

Recent mathematical approaches to solving these problems

Identification of open problems in shock wave theory

Abstract

Shock waves are steep wave fronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several longstanding shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches, and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection-diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.