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

TL;DR

This paper applies the Nash-Moser theorem to construct solutions for the vacuum boundary problem of gaseous stars, extending previous methods to cases where the adiabatic exponent's reciprocal is not an integer but large.

Contribution

It extends the application of the Nash-Moser theorem to cases with non-integer reciprocal of the adiabatic exponent, broadening the scope of solutions for gaseous star models.

Findings

Successfully constructed solutions for non-integer reciprocal cases.

Extended Nash-Moser theorem application to relativistic and non-relativistic models.

Provided a framework for solutions near equilibrium states.

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.