The Mathematics of Shock Reflection-Diffraction and von Neumann’s Conjectures

As part of our series of research articles deliberately focusing on the rigour and complexity of mathematics and its problems, Oxford Mathematician Gui-Qiang G Chen discusses his work on the Mathematics of Shock Reflection-Diffraction.

Shock waves are fundamental in nature, especially in high-speed fluid flows. Shocks are often generated by supersonic or near-sonic aircraft, explosions, solar wind, and other natural processes. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws - nonlinear partial differential equations (PDEs) of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns as conjectured by von Neumann (1943), it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs—mixed type, free boundaries, and corner singularities—that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science.

Oxford mathematician Gui-Qiang G. Chen and his collaborator Mikhail Feldman (University of Wisconsin-Madison) have introduced new ideas and developed techniques for solving fundamental open problems for multidimensional (M-D) shock reflection-diffraction and related free boundary problems for nonlinear conservation laws of mixed hyperbolic-elliptic type in a series of their papers. In particular, in their Annals paper, they developed the first mathematical approach to the global problem of shock reflection-diffraction by wedges and employed the approach to solve rigorously the problem with large-angle wedges for potential flow through careful mathematical analysis. This paper was awarded the Analysis of Partial Differential Equations Prize in 2011 by the Society for Industrial and Applied Mathematics. 

In the last five years, further significant advances have been made, including their complete solution to von Neumann’s sonic conjecture and detachment conjecture for potential flow. These are reported in their forthcoming research monograph published in the Princeton Series in Annals of Mathematics Studies. This monograph offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of their original mathematical proofs of von Neumann's conjectures, and a collection of related results and new techniques in the analysis of PDEs, as well as a set of fundamental open problems for further development. The approaches and techniques that Chen and his collaborators have developed will be useful in solving nonlinear problems with similar difficulties and open up new research directions.