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

Part of: Hyperbolic equations and systems Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  03 October 2025

Geng Chen ,
Houbin Guo  and
Yanbo Hu Open the ORCID record for Yanbo Hu [Opens in a new window]
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
*
e-mail: yanbo.hu@hotmail.com
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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.

Keywords

Relativistic Euler equationsglobal smooth solutionsingularitycharacteristic methodradial symmetry

MSC classification

Primary: 35Q31: Euler equations 35L65: Conservation laws 35Q75: PDEs in connection with relativity and gravitational theory 35A09: Classical solutions 35B44: Blow-up

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 32
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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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