Global Entropy Solutions and Newtonian Limit for the Relativistic Euler Equations

Gui-qiang G. Chen, Matthew R. I. Schrecker

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Abstract

We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on a careful entropy analysis, we establish the compactness framework for sequences of both exact solutions and approximate solutions of the relativistic Euler equations. Then we construct approximate solutions via the vanishing viscosity method and employ our compactness framework to deduce the global-in-time existence of entropy solutions. The compactness of the solution operator is also established. Finally, we apply our techniques to establish the convergence of the Newtonian limit from the entropy solutions of the relativistic Euler equations to the classical Euler equations.
Original languageEnglish
Article number10
Number of pages53
JournalAnnals of PDE
Volume8
Issue number1
DOIs
Publication statusPublished - 12 May 2022

Bibliographical note

Acknowledgements:
The research of Gui-Qiang G. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/L015811/1, EP/V008854, and EP/V051121/1, and the Royal Society–Wolfson Research Merit Award WM090014 (UK). The research of Matthew Schrecker was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/L015811/1. This paper is a continuation of the program initiated by Gui-Qiang G. Chen, Philippe LeFloch, and Yachun Li in [4,5,6,7]. The authors thank Professors Ph. LeFloch and Yachun Li for their inputs and helpful discussions.

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