Kyoto Journal of Mathematics
- Kyoto J. Math.
- Volume 56, Number 2 (2016), 243-282.
On spherically symmetric solutions of the Einstein–Euler equations
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.
Kyoto J. Math., Volume 56, Number 2 (2016), 243-282.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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