Shock Reflection-Diffraction, von Neumann's Conjectures and Nonlinear Equations of Mixed Type

TL;DR

This paper reviews the historical and mathematical study of shock reflection-diffraction phenomena, focusing on von Neumann's conjectures and recent advances in understanding the solutions to related nonlinear mixed-type equations.

Contribution

It formulates the shock reflection-diffraction problem as a boundary value problem for nonlinear conservation laws and discusses progress on von Neumann's conjectures and solution regularity.

Findings

Progress on solving von Neumann's sonic and detachment conjectures

Existence and stability of global regular shock reflection configurations

Development of a mathematical theory for shock reflection-diffraction

Abstract

Shock waves are fundamental in nature. One of the most fundamental problems in fluid mechanics is shock reflection-diffraction by wedges. The complexity of reflection-diffraction configurations was first reported by Ernst Mach in 1878. The problems remained dormant until the 1940s when John von Neumann, as well as other mathematical/experimental scientists, began extensive research into all aspects of shock reflection-diffraction phenomena. In this paper we start with shock reflection-diffraction phenomena and historic perspectives, their fundamental scientific issues and theoretical roles in the mathematical theory of hyperbolic systems of conservation laws. Then we present how the global shock reflection-diffraction problem can be formulated as a boundary value problem in an unbounded domain for nonlinear conservation laws of mixed hyperbolic-elliptic type, and describe the von Neumann conjectures: the sonic conjecture and the detachment conjecture. Finally we discuss some recent developments in solving the von Neumann conjectures and establishing a mathematical theory of shock reflection-diffraction, including the existence, regularity, and stability of global regular configurations of shock reflection-diffraction by wedges.