Regular Shock Reflection-diffraction Problem for the Euler Flow

Project: Research

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Description

The aim of this project is to develop the rigorous mathematical theory on the regular shock reflection-diffraction problem for the Euler flow. In particular, it will mainly focus on the global existence of solutions of the regular shock reflection-diffraction problem governed by the two dimensional compressible Euler equations, and other related problems. The global existence of solutions of the regular shock reflection-diffraction problem for the Euler flow is a longstanding open problem in the mathematical fluid dynamics, due to many serious analytical difficulties, such as nonlinear partial differential equations of composite-mixed hyperbolic-elliptic type, free boundary problems for nonlinear degenerate partial differential equations with corner singularities, degeneracy of the transport equation of the vorticity with low regularity, etc. Moreover, the shock reflection-diffraction problem is one of the typical and core two-dimensional Riemann problems, whose solutions are local building blocks and large-time asymptotic states of the general entropy solutions of multidimensional hyperbolic conservation laws. In this project, we will first try to establish the global existence of regular shock reflection-diffraction solutions when the wedge angle is nearp/2. Next, we will consider the convexity of the diffracted transonic shock, and other related problems. Any techniques and approaches developed for solving the problems in this project will be helpful to both other problems with similar difficulties and the mathematical theory on the multi-dimensional conservation laws. There having been a lot of literatures on the global existence and geometrical structures of solutions of the potential flow equation in a self-similar coordinates, as well as the local structure of solutions of the Mach reflections ( cf. [1, 2, 4, 7-11, 14-17, 19, 21, 23, 29, 32, 35, 37, 39, 42]). However, there is no rigorous analysis on the global shock reflection-diffraction solutions governed by the isentropic/full Euler equations up to now. Due to the importance of this topic in nonlinear analysis and aerodynamics, it is valuable for us to study the shock reflection diffraction problem by working on the objectives proposed in this project based on the techniques developed for the potential flow equation. We have been working on the related problems for several years. We proved the global existence and optimal regularity of solutions for shock diffraction problem to the nonlinear wave system in [5]; loss of regularity of solutions of shock diffraction problem to the potential flow equation in [10]; convexity of transonic shocks in self-similar coordinates in [11]; uniqueness and structural stability of solutions of regular shock reflection-diffraction problem for the potential flow in [12]. In this project, we will mainly focus on a further development of the technics of proving the obtained results, to study the mentioned objectives.

Detail(s)

Project number9043179
Grant typeGRF
StatusNot started
Effective start/end date1/01/22 → …