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Global solution to a three-dimensional spherical piston problem for the relativistic Euler equations

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  23 September 2020

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: laigeng@shu.edu.cn
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Abstract

The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we consider a ‘spherical piston’ problem for the relativistic Euler equations, which describes the wave motion produced by a sphere expanding into an infinite surrounding medium. We use the reflected characteristics method to construct a global piecewise smooth solution with a single shock of this spherical piston problem, provided that the speed of the sphere is a small perturbation of a constant speed.

Keywords

Relativistic Euler equationsspherical piston problemshock wavecharacteristic

MSC classification

Primary: 35L65: Conservation laws
Secondary: 35L60: Nonlinear first-order hyperbolic equations 35L67: Shocks and singularities

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 1 - 26
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Chen, G. Q., Chen, S. X., Wang, D. H. & Wang, Z. J. (2005) A multidimensional piston problem for the Euler equations for compressible flow. Discrete Contin. Dyn. Syst. 13, 361–383.CrossRefGoogle Scholar
Chen, G. Q. & Li, Y. C. (2004) Stability of Riemann solutions with large oscillation for the relativistic Euler equations. J. Differ. Equ. 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
Chen, J. (1995) Conservation laws for the relativistic p-system. Comm. Part. Differ. Equ. 20, 1605–1646.CrossRefGoogle Scholar
Chen, J. (1997) Conservation laws for relativistic fluid dynamics. Arch. Ration. Mech. Anal. 139, 377–398.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
Courant, R. & Friedrichs, K. O. (1948) Supersonic Flow and Shock Waves, Interscience, New York.Google Scholar
Cui, D. C. & Yin, H. C. (2007) Global supersonic conic shock wave for the steady supersonic flow past a cone: isothermal case. Pacific J. Math. 233, 257–289.CrossRefGoogle Scholar
Cui, D. C. & Yin, H. C. (2009) Global supersonic conic shock wave for the steady supersonic flow past a cone: polytropic gas. J. Differ. Equ. 246, 641–669.CrossRefGoogle 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. In: Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proceedings in Mathematics & Statistics, Vol. 49, Springer, Heidelberg, pp. 161–191.CrossRefGoogle Scholar
Hsu, C. H., Lin, S. S. & Makino, T. (2001) On the relativistic Euler equation. Methods Appl. Anal. 8, 159–208.CrossRefGoogle Scholar
Hsu, C. H., Lin, S. S. & Makino, T. (2004) On spherically symmetric solutions of the relativistic Euler equation. J. Differ. Equ. 201, 1–24 CrossRefGoogle Scholar
Lai, G. (2020) Self-similar solutions of the radially-symmetric relativistic Euler equations. Eur. J. Appl. Math. doi: 10.1017/S0956792519000317.CrossRefGoogle Scholar
Landau, L. D. & Lifschitz, E. M. (1987) Fluid Mechanics, Pergamon, Oxford.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
Li, T. T. (1994) Global Classical Solutions for Quasilinear Hyperbolic System, John Wiley and Sons, Paris.Google Scholar
Martí, 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
Smoller, J. & Temple, B. (1993) Global solutions of the relativistic Euler equations. Comm. Math. Phys. 156, 67–99.CrossRefGoogle Scholar
Steinhardt, P. J. (1982) Relativistic detonation waves and bubble growth in false vacuum decay. Phys. Rev. D 25, 2074–2085.CrossRefGoogle Scholar
Taub, A. H. (1948) Relativistic Rankine-Hugoniot Equations. Physical Rev. 74, 328–334.CrossRefGoogle Scholar
Taylor, G. I. (1946) The air wave surrounding an expanding sphere. Proc. R. Soc. London 186, 273–292.Google ScholarPubMed
Wissman, B. D. (2011) Global Solutions to the Ultra-Relativistic Euler Equations. Comm. Math. Phys. 306, 831–851.CrossRefGoogle Scholar
Yin, H. C. (2006) Global existence of a shock for the supersonic flow past a curved wedge. Acta Math. Sinica 22, 1425–1432.CrossRefGoogle Scholar