Kyoto Journal of Mathematics

On spherically symmetric solutions of the Einstein–Euler equations

Tetu Makino

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Abstract

We construct spherically symmetric solutions to the Einstein–Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein–Euler equations are given by the Tolman–Oppenheimer–Volkoff equations, and time-periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying a so-called physical boundary condition at the free boundary with the vacuum will be constructed using the Nash–Moser theorem. This work also can be considered as a touchstone in order to estimate the universality of the method which was originally developed for the nonrelativistic Euler–Poisson equations.

Article information

Source
Kyoto J. Math., Volume 56, Number 2 (2016), 243-282.

Dates
Received: 14 October 2014
Revised: 6 February 2015
Accepted: 13 February 2015
First available in Project Euclid: 10 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1462901079

Digital Object Identifier
doi:10.1215/21562261-3478880

Mathematical Reviews number (MathSciNet)
MR3500842

Zentralblatt MATH identifier
1351.35220

Subjects
Primary: 35L05: Wave equation 35L52: Initial value problems for second-order hyperbolic systems 35L57: Initial-boundary value problems for higher-order hyperbolic systems 35L70: Nonlinear second-order hyperbolic equations
Secondary: 76L10 76N15: Gas dynamics, general 83C05: Einstein's equations (general structure, canonical formalism, Cauchy problems) 85A30: Hydrodynamic and hydromagnetic problems [See also 76Y05]

Keywords
Einstein equations spherically symmetric solutions vacuum boundary Nash–Moser theorem

Citation

Makino, Tetu. On spherically symmetric solutions of the Einstein–Euler equations. Kyoto J. Math. 56 (2016), no. 2, 243--282. doi:10.1215/21562261-3478880. https://projecteuclid.org/euclid.kjm/1462901079


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