[Plenary Lecture] Multidimensional Riemann Problems and Hyperbolic Conservation Laws

Speaker: Gui-Qiang G. Chen, Oxford University

Title: Multidimensional Riemann Problems and Hyperbolic Conservation Laws

Abstract: In this talk, we first discuss the underlying connections between multidimensional Riemann problems—focusing on their global patterns and structures—and general entropy solutions of nonlinear hyperbolic systems of conservation laws. We then present recent progress in the rigorous analysis of several longstanding two-dimensional Riemann problems, both initial and lateral, involving transonic shock waves for the Euler equations of potential flow. In particular, we focus on the four-shock Riemann problem for the Euler equations for potential flow, as a representative example, to illustrate how these problems can be reformulated and solved as free boundary problems, with transonic shock waves serving as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic–hyperbolic type. We further address several physically significant lateral Riemann problems, including Prandtl’s reflection problem and von Neumann’s shock reflection-diffraction problem. Moreover, we present different regularity properties of Riemann solutions to the compressible Euler equations for both potential and isentropic flows. We also discuss additional multidimensional Riemann problems and related shock-wave problems for nonlinear hyperbolic systems of conservation laws, if time permits.