Potential theory for shock reflection by a large-angle wedge

When a plane shock hits a wedge head on, it experiences a reflection, and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of reflected shocks may occur, including regular and Mach reflection. However, most fundamental issues for shock reflection phenomena have not been understood, such as the transition among the different patterns of shock reflection; therefore, it is essential to establish a global existence and stability theory for shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and stability of solutions to shock reflection, especially for potential flow, which has widely been used in aerodynamics. The theoretical problems involve several challenging difficulties in the analysis of nonlinear partial differential equations including elliptic-hyperbolic mixed type, free-boundary problems, and corner singularity, especially when an elliptic degenerate curve meets a free boundary. Here we develop a potential theory to overcome these difficulties and to establish the global existence and stability of solutions to shock reflection by a large-angle wedge for potential flow. The techniques and ideas developed will be useful to other nonlinear problems involving similar difficulties.