Multi-Dimensional Steady Contact Discontinuity

The aim of this project is to develop rigorous mathematical theory on the multi-dimensional steady contact discontinuity governed by the compressible Euler equations. In particular, it mainly focus on the global existence of the three dimensional contact discontinuity for a supersonic flow over a convex cornered wedge, the stability of a transonic contact discontinuity in an axisymmetric nozzle and other related problems. As said in Courant-Friedrichs in [24], compressible fluids exhibit conspicuous nonlinear phenomena, such as Mach shock configurations and jet flows. All these phenomena can be formulated by elementary waves and their interactions. As one of the elementary waves, contact discontinuity is of great interest to be investigated rigorously due to the importance in applications as well as the complexity. In this project, we will study the stability of the contact discontinuity by working on several important steady structures. Precisely, we will first try to establish the stability of a three dimensional transonic contact discontinuity in an axisymmetric nozzle. Next, we will consider global existence of a three dimensional transonic shock and a contact discontinuity over a convex cornered wedge and other related problems. Mathematically, these problems can be formulated into a free boundary value problem governed by nonlinear equations of hyperbolic-elliptic mixed-type. The crucial difficulty comes from the fact that the contact discontinuity is a free interface, on both sides of which the states are unknown, and it is characteristic from the hypersonic side, especially coupled with other difficulties such as nonlinearity, corner singularity, etc. Up to now, there have been lots of literatures on the existence and stability of two-dimensional steady contact discontinuities governed by the compressible Euler equations. It is natural to continue the study of the steady contact continuities in three dimensions. However the rigorous mathematical theory for such problems is far away from being satisfied. We believe the ideas and techniques developed in this project would be helpful for the mathematical theory on the multi-dimensional conservation laws. Other mathematicians in the fields of partial differential equations and mathematical fluid dynamics would be interested in this work too . We have been working on the related problems for several years. We established the stability of a supersonic contact discontinuity in a finitely long nozzle in [30]; the stability of a transonic contact discontinuity in a finitely long nozzle in [31]; the global existence and uniqueness of a transonic shock with a jet over a convex cornered wedge in [36]; the local stability of a three dimensional rarefaction wave and a contact discontinuity over a plane wedge in [37]; the global existence of axisymmetric Euler flows in an infinitely long nozzle with non-trivial swirl in [25]; the global existence and uniqueness of two dimensional steady contact discontinuities in an infinitely long nozzle with large vorticity in [9]; the stability of a three dimensional weak transonic shock over a wedge in [5]. In this project, we will mainly focus on further development of the technics used in the works above, to study the mentioned objectives.