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Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

  • Yu Zhang EMAIL logo and Yanyan Zhang
Published/Copyright: March 5, 2021

Abstract

We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.

Mathematics Subject Classification: 35L65; 35L67; 76N10; 76N15

Corresponding author: Yu Zhang, Department of Mathematics, Yunnan Normal University, Kunming, 650500, PR China, E-mail:

Award Identifier / Grant number: 11501488

Funding source: Nan Hu Young Scholar Supporting Program of XYNU

Funding source: Scientific Research Foundation Project of Yunnan Education Department

Award Identifier / Grant number: 2018JS150

Funding source: Yunnan Applied Basic Research Projects

Award Identifier / Grant number: 2018FD015

Funding source: the PhD research startup foundation of Yunnan Normal UniversityYunnan Normal UniversityXinyang Normal University

Award Identifier / Grant number: 2016zb012

Award Identifier / Grant number: Unassigned

Award Identifier / Grant number: Unassigned

Acknowledgements

The authors specially thank the referee for many valuable suggestions to revise this paper.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work is supported by National Natural Science Foundation of China (No. 11501488), Yunnan Applied Basic Research Projects (No. 2018FD015), Scientific Research Foundation Project of Yunnan Education Department (No. 2018JS150), the PhD research startup foundation of Yunnan Normal University (No. 2016zb012) and Nan Hu Young Scholar Supporting Program of XYNU.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2020-02-24
Revised: 2020-10-18
Accepted: 2021-02-07
Published Online: 2021-03-05
Published in Print: 2022-10-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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