Shock Reflection-Diffraction and Nonlinear Partial Differential Equations of Mixed Type

We present our recent results on the existence and regularity of shock reflectiondiffraction configurations by a wedge up to the sonic wedge angle, which is the von Neumann sonic conjecture. The problem is first formulated as a boundary value problem for a second-order nonlinear partial differential equation of mixed hyperbolic-elliptic type in an unbounded domain. Then the boundary value problem is reduced into a one-phase free boundary problem for a nonlinear second-order degenerate elliptic equation with the free boundary meeting the degenerate curve on which the ellipticity of the equation fails. The key steps to achieve the results are the a priori estimates of admissible solutions of the shock reflection-diffraction problem, which yield the existence theory. 1. Shock Reflection-Diffraction Problems Shock reflection-diffraction by a straight-sided wedge is one of the most fundamental multidimensional shock wave phenomena. When a plane shock hits the wedge head on, a self-similar shock of reflection-diffraction moves outward as the original shock moves forward in time. Such problems not only arise in many important physical situations, but also are fundamental in the mathematical theory of multidimensional conservation laws since their solutions are building blocks and asymptotic attractors of general solutions to the multidimensional Euler equations for compressible fluids (cf. CourantFriedrichs [9], von Neumann [27, 28], Glimm-Majda [12], and Morawetz [24]). The complexity of reflection-diffraction configurations was first reported by Ernst Mach [23] in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of reflected shocks may occur, including regular and Mach reflection (cf. [2, 11, 12, 13, 14, 15, 16, 17, 19, 21, 24, 26, 27, 28]). Most of the fundamental issues for shock reflection-diffraction have not been understood, such as the transition between different patterns, especially for the potential flow equation used widely in aerodynamics. 1991 Mathematics Subject Classification. Primary: 35M12, 3502, 35L65, 35L67, 35J70, 35R35, 76H05, 76N10; Secondary: 35M13, 76L05.