Kyoto Journal of Mathematics

On spherically symmetric solutions of the Einstein–Euler equations

Tetu Makino

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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]

Einstein equations spherically symmetric solutions vacuum boundary Nash–Moser theorem


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.

Export citation


  • [1] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 3rd ed., Wiley, New York, 1978.
  • [2] U. Brauer and L. Karp, Local existence of solutions of self gravitating relativistic perfect fluids, Comm. Math. Phys. 325 (2014), 105–141.
  • [3] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65–222.
  • [4] M. Hukuhara, T. Kimura, and T. Matuda, Equations différentielles ordinaires du premier ordre dans le champ complexe, Publ. Math. Soc. Japan, Tokyo, 1961.
  • [5] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2, 4th ed., Pergamon Press, Oxford, 1975.
  • [6] T. Makino, “On a local existence theorem for the evolution equation of gaseous stars” in Patterns and Waves, Stud. Math. Appl. 18, North-Holland, Amsterdam, 1986, 459–479.
  • [7] T. Makino, On spherically symmetric stellar models in general relativity, Kyoto J. Math. 38 (1998), 55–69.
  • [8] T. Makino, On spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations, Osaka J. Math. 52 (2015), 545–580.
  • [9] T. Makino, On spherically symmetric motions of the atmosphere surrounding a planet governed by the compressible Euler equations, Funkcial. Ekvac. 58 (2015), 43–85.
  • [10] C. W. Misner and D. H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. (2) 136 (1964), B571–B576.
  • [11] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco, Calif., 1970.
  • [12] J. P. Oppenheimer and G. M. Volkoff, On massive neutron cores, Phys. Rev. 55 (1939), 374–381.
  • [13] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
  • [14] A. D. Rendall and B. G. Schmidt, Existence and properties of spherically symmetric static fluid bodies with a given equation of state, Class. Quantum Gravity 8 (1991), 985–1000.
  • [15] Ya. B. Zeldovich and I. D. Novikov, Relativistic Astrophysics, 1: Stars and Relativity, Univ. Chicago Press, Chicago, 1971.