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

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.