An application of the Nash-Moser theorem to the vacuum boundary problem of gaseous stars

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/(γ−1) is an integer, where γ 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/(γ−1) is not an integer but sufficiently large.