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Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems

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Abstract

We are concerned with global solutions of multidimensional (M-D) Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous analysis of two-dimensional (2-D) Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations. In particular, we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

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Acknowledgements

This paper is dedicated to Professor Tong Zhang (Tung Chang) on the occasion of his 90th birthday, who has been one of the pioneers and main contributors in the analysis of the 2-D Riemann problems; see for example [8,9,10,11,12, 59, 80, 92,93,94] and the references cited therein. The research of Gui-Qiang G. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1, EP/V008854/1, EP/V051121/1, and the Royal Society-Wolfson Research Merit Award WM090014.

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    Gui-Qiang G. Chen

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Chen, GQ.G. Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems. Commun. Appl. Math. Comput. 5, 1015–1052 (2023). https://doi.org/10.1007/s42967-022-00210-4

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