Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T21:41:50.263Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  23 September 2020

GENG LAI*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. China email: [email protected]

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.

Type
Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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.)

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, 361383.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, 332353.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, 903926.CrossRefGoogle Scholar
Chen, J. (1995) Conservation laws for the relativistic p-system. Comm. Part. Differ. Equ. 20, 16051646.CrossRefGoogle Scholar
Chen, J. (1997) Conservation laws for relativistic fluid dynamics. Arch. Ration. Mech. Anal. 139, 377398.CrossRefGoogle Scholar
Cheng, H. J. & Yang, H. C. (2011) Riemann problem for the relativistic Chaplygin Euler equations. J. Math. Anal. Appl. 381, 1726.CrossRefGoogle Scholar
Cheng, H. J. & Yang, H. C. (2012) Riemann problem for the isentropic relativistic Chaplygin Euler equations. Z. Angew. Math. Phys. 63, 429440.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, 257289.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, 641669.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, 101121.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. 161191.CrossRefGoogle Scholar
Hsu, C. H., Lin, S. S. & Makino, T. (2001) On the relativistic Euler equation. Methods Appl. Anal. 8, 159208.CrossRefGoogle Scholar
Hsu, C. H., Lin, S. S. & Makino, T. (2004) On spherically symmetric solutions of the relativistic Euler equation. J. Differ. Equ. 201, 124 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, 239253.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, 763778.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, 317333.CrossRefGoogle Scholar
Mizohata, K. (1997) Global solution to the relativistic Euler equation with spherical symmetry. J. Indust. Apol. Math. 14, 125157.Google Scholar
Smoller, J. & Temple, B. (1993) Global solutions of the relativistic Euler equations. Comm. Math. Phys. 156, 6799.CrossRefGoogle Scholar
Steinhardt, P. J. (1982) Relativistic detonation waves and bubble growth in false vacuum decay. Phys. Rev. D 25, 20742085.CrossRefGoogle Scholar
Taub, A. H. (1948) Relativistic Rankine-Hugoniot Equations. Physical Rev. 74, 328334.CrossRefGoogle Scholar
Taylor, G. I. (1946) The air wave surrounding an expanding sphere. Proc. R. Soc. London 186, 273292.Google ScholarPubMed
Wissman, B. D. (2011) Global Solutions to the Ultra-Relativistic Euler Equations. Comm. Math. Phys. 306, 831851.CrossRefGoogle Scholar
Yin, H. C. (2006) Global existence of a shock for the supersonic flow past a curved wedge. Acta Math. Sinica 22, 14251432.CrossRefGoogle Scholar