2013/11/21 by Gui‐Qiang Chen, Gui-Qiang G. Chen, Chen, Gui-Qiang G. +2
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #math.AP
paper · pdf · doi:10.48550/arxiv.1311.5596
10 pages, 3 figures; Proceedings of the 6th International Congress of Chinese Mathematicians, 2013 (to appear)
arxiv created 2013/11/21 · openalex publication_date 2013/11/21 · arxiv updated 2013/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.