Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T07:39:20.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-similar solutions of the radially symmetric relativistic Euler equations

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  04 November 2019

GENG LAI Open the ORCID record for GENG LAI [Opens in a new window]
GENG LAI*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.

Keywords

Relativistic Euler equationsradial symmetryRiemann problemself-similar solution

MSC classification

Primary: 35L65: Conservation laws
Secondary: 35L60: Nonlinear first-order hyperbolic equations 35L67: Shocks and singularities
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 6 , December 2020 , pp. 919 - 949
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

This work was partially supported by NSF of China 11301326 and the grant of ‘Shanghai Altitude Discipline’.

References

Chen, G. Q. & Li, Y. C. (2004) Stability of Riemann solutions with large oscillation for the relativistic Euler equations. J. Differential Equations, 202, 332–353.CrossRefGoogle Scholar
Chen, G. Q. & Li, Y. C. (2004) Relativistic Euler equations for isentropic fluids: stability of Riemann solutions with large oscillation. Z. Angew. Math. Phys. 55, 903–926.CrossRefGoogle Scholar
Cheng, H. J. & Yang, H. C. (2011). Riemann problem for the relativistic Chaplygin Euler equations. J. Math. Anal. Appl. 381, 17–26.CrossRefGoogle Scholar
Cheng, H. J. & Yang, H. C. (2012). Riemann problem for the isentropic relativistic Chaplygin Euler equations. Z. Angew. Math. Phys. 63, 429–440.CrossRefGoogle Scholar
Chen, J. (1995). Conservation laws for the relativistic p-system. Comm. Partial Differential Equations, 20, 1605–1646.CrossRefGoogle Scholar
Chen, J. (1997). Conservation laws for relativistic fluid dynamics. Arch. Ration. Mech. Anal. 139, 377–398.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K. O. (1948) Supersonic Flow and Shock Waves. Interscience, New York.Google Scholar
Ding, M. & Li, Y. C. (2013) Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations. Z. Angew. Math. Phys. 64, 101–121.CrossRefGoogle Scholar
Ding, M. & Li, Y. C. (2014) An overview of piston problems in fluid dynamics. Hyperbolic conservation laws and related analysis with applications. Springer Proc. Math. Stat. 49, 161–191.Google Scholar
Hsu, C. H., Lin, S. S. & Makino, T. (2001) On the relativistic Euler equation. Methods Appl. Anal., 8, 159–208.Google Scholar
Hsu, C. H., Lin, S. S. & Makino, T. (2004) On spherically symmetric solutions of the relativistic Euler equation. J. Differential Equations, 201, 1–24CrossRefGoogle Scholar
Jenssen, H. K. (2011) On radially symmetric solutions to conservation laws. Nonlinear conservation laws and applications. IMA Vol. Math. Appl., 153, 331–351.Google Scholar
Landau, L. D. & Lifschitz, E. M. (1987) Fluid Mechanics, Pergamon, Oxford.Google Scholar
Li, T. T. & Qin, T. H. (2005). Physics and Partial Differential Equations (in Chinese), 2nd ed. Higher Education Press, Beijing.Google Scholar
Li, Y. C., Feng, D. M. & Wang, Z. J. (2005) Global entropy solutions to the relativistic Euler equations for a class of large initial data. Z. Angew. Math. Phys. 56, 239–253.CrossRefGoogle Scholar
Li, Y. C. & Shi, Q. F. (2005) Global existence of the entropy solutions to the isentropic relativistic Euler equations. Commun. Pure Appl. Anal., 4, 763–778.CrossRefGoogle Scholar
Mart, J. M. & Müller, E. (1994) The analytical solution of the Riemann problem in relativistic hydrodynamics. J. Fluid Mech. 258, 317–333.CrossRefGoogle Scholar
Mizohata, K. (1997) Global solution to the relativistic Euler equation with spherical symmetry. J. Indust. Apol. Math., 14, 125–157.Google Scholar
Peng, C. C. & Lien, W. C. (2012). Self-similar solutions of the Euler equations with spherical symmetry. Nonlinear Analysis, 75, 6370–6378.CrossRefGoogle Scholar
Steinhardt, P. J. (1982). Relativistic detonation waves and bubble growth in false vacuum decay. Physical Review D., 25, 2074–2085.CrossRefGoogle Scholar
Smoller, J. & Temple, B. (1993) Global Solutions of The Relativistic Euler Equations. Comm. Math. Phys., 156, 67–99.CrossRefGoogle Scholar
Taylor, G. I. (1946) The air wave surrounding an expanding sphere. Proceedings of the Royal Society of London, 186, 273–292.Google ScholarPubMed
Taub, A. H. (1948) Relativistic Rankine-Hugoniot equations. Physical Rev., 74, 328–334.CrossRefGoogle Scholar
Wissman, B. D. (2011) Global solutions to the ultra-relativistic Euler equations. Comm. Math. Phys., 306, 831–851.CrossRefGoogle Scholar
Zhang, T. & Zheng, Y. X. (1998) Axisymmetric solutions of the Euler equations for polytropic gases. Arch. Ration. Mech. Anal. 142, 253–279.CrossRefGoogle Scholar
Zheng, Y. X. (2001) Systems of Conservation Laws: 2-D Riemann Problems. 38 PNLDE, Bikhäuser, Boston.CrossRefGoogle Scholar