Gui-Qiang G. Chen, Oxford Centre for Nonlinear PDE,
University of Oxford
Shock Reflection-Diffraction, Free Boundary Problems, and
Nonlinear PDEs of Mixed Hyperbolic-Elliptic Type
Shock waves are steep fronts that propagate in the compressible fluids when the convective motion dominates the diffusion, which are fundamental in nature. 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 may occur, including regular, Mach, and von Neumann reflection. However, most fundamental issues have not been understood, including the transition of different patterns of shock reflection-diffraction. Therefore, it becomes essential to establish a mathematical theory on the existence, stability, and regularity of global configurations of shock reflection-diffraction, especially for potential flow which has widely been used in aerodynamics.
In this talk we will start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we will describe how the global shock reflection-diffraction problems can be formulated as free boundary problems for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type. Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including the existence, stability, and regularity of global regular configurations of shock reflection-diffraction by wedges. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities. Further trends and open problems in this direction will also be addressed.