Shock Reflection-Diffraction Phenomena and Multidimensional Conservation Laws

TL;DR

This paper explores shock reflection-diffraction phenomena, their mathematical modeling as free boundary problems, and recent advances in proving existence, stability, and regularity of solutions within the context of multidimensional conservation laws.

Contribution

It formulates the shock reflection-diffraction problem as a free boundary problem and discusses new analytical techniques for establishing solution properties.

Findings

Existence of global regular reflection-diffraction solutions

Stability and regularity results for shock reflection configurations

Development of methods to handle free boundary and corner singularities

Abstract

When a plane shock hits a wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of reflection-diffraction configurations was first reported by Ernst Mach in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection-diffraction configurations may occur, including regular reflection and Mach reflection. In this paper we start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles as building blocks and asymptotic attractors of general solutions in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we describe how the global problem of shock reflection-diffraction by a wedge can be formulated as a free boundary problem for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type. Finally we discuss some recent developments in attacking the shock reflection-diffraction problem, including the existence, stability, and regularity of global regular reflection-diffraction solutions. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. Further trends and open problems in this direction are also addressed.