Zhouping Xin

Abstract:   The mathematical theory of systems of multi-dimensional hyperbolic conservations remains a great challenge to the field of nonlinear partial differential eqations. Some of the main difficulties in dealing with such a system are the system is mixed type and may change types so that the classical approaches fail to yield the global well-posedness theory. This is so even for the important compressible Euler systems in gas dynamics. In this talk, I will illustrate these by the example of the transonic steady compressible flows in multi-dimensional ducts with varaible sections. In particular, I will discuss some recent progress on the transonic shock problem in a class de Laval nozzles due to Courant-Friedrichs and exsitence of smooth subsonic-sonic and transonic flows in such nozzles.