An Application of the Nash-Moser Theorem to the Vacuum Boundary Problem of Gaseous Stars

Abstract

We have been studying spherically symmetric motions of gaseous stars with physical vacuum boundary governed either by the Euler-Poisson equations in the non-relativistic theory or by the Einstein-Euler equations in the relativistic theory. The problems are to construct solutions whose first approximations are small time-periodic solutions to the linearized problem at an equilibrium and to construct solutions to the Cauchy problem near an equilibrium. These problems can be solved when $1/(\gamma -1)$ is an integer, where $\gamma$ is the adiabatic exponent of the gas near the vacuum, by the formulation by R. Hamilton of the Nash-Moser theorem. We discuss on an application of the formulation by J. T. Schwartz of the Nash-Moser theorem to the case in which $1/(\gamma -1)$ is not an integer but sufficiently large.

Typos corrected, and References are up-dated. No substantial changes

References in corpus (2)

• On the Tolman-Oppenheimer-Volkoff-de Sitter equation
• On a certain type of nonlinear hyperbolic equations derived from astrophysical problems

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• Recent developments in mathematical aspects of relativistic fluids
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• On the asymptotic behavior of static perfect fluids
• Dynamical relativistic liquid bodies I: constraint propagation
• A priori estimates for a relativistic liquid with free surface boundary
• On spherically symmetric solutions of the Einstein-Euler-de Sitter equations