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Global smooth solutions and singularity formation for the relativistic Euler equations with radial symmetry

Published online by Cambridge University Press:  03 October 2025

Geng Chen
Affiliation:
The University of Kansas , United States e-mail: gengchen@ku.edu
Houbin Guo
Affiliation:
South China University of Technology , Guangzhou, China e-mail: ghbHZNU123@yeah.net
Yanbo Hu*
Affiliation:
Zhejiang University of Science and Technology , China

Abstract

In this article, we consider the global-in-time existence and singularity formation of smooth solutions for the radially symmetric relativistic Euler equations of polytropic gases. We introduce the rarefaction/compression character variables for the supersonic expanding wave with relativity and derive their Riccati type equations to establish a series of priori estimates of solutions by the characteristic method and the invariant domain idea. It is verified that, for rarefactive initial data with vacuum at the origin, smooth solutions will exist globally. On the other hand, the smooth solution develops a singularity in finite time when the initial data are compressed and include strong compression somewhere.

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Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

G.C. was partially supported by the National Science Foundation (DMS-2306258). Y.H. was partially supported by the National Natural Science Foundation of China (12171130) and the Natural Science Foundation of Zhejiang Province of China (LMS25A010014).

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