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Spatially Periodic Solutions in Relativistic Isentropic Gas Dynamics

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Abstract

We consider the initial value problem, with periodic initial data, for the Euler equations in relativistic isentropic gas dynamics, for ideal polytropic gases which obey a constitutive equation, relating pressure p and density ρ, p=κ2ργ, with γ≥1, 0<κ<c, where c is the speed of light. Global existence of periodic entropy solutions for initial data sufficiently close to a constant state follows from a celebrated result of Glimm and Lax (1970). We prove that given any periodic initial data of locally bounded total variation, satisfying the physical restrictions ||v0||∞<c, where v is the gas velocity, there exists a globally defined spatially periodic entropy solution for the Cauchy problem, if 1≤γ<γ0, for some γ0>1, depending on the initial bounds. The solution decays in L loc 1 to its mean value as t→∞.

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Authors and Affiliations

  1. Instituto de Matemática, Pura e Aplicada-IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro RJ, Brasil

    Hermano Frid

  2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois, 60208-2730, USA

    Mikhail Perepelitsa

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  1. Hermano Frid
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  2. Mikhail Perepelitsa
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Correspondence to Hermano Frid.

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Communicated by P. Constantin

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Frid, H., Perepelitsa, M. Spatially Periodic Solutions in Relativistic Isentropic Gas Dynamics. Commun. Math. Phys. 250, 335–370 (2004). https://doi.org/10.1007/s00220-004-1148-x

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  • DOI: https://doi.org/10.1007/s00220-004-1148-x

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