## Kyoto Journal of Mathematics

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

### On spherically symmetric solutions of the Einstein–Euler equations

#### 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