Well-Posedness of the Einstein–Euler System in Asymptotically Flat Spacetimes

We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein–Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for elliptic and hyperbolic equations in these spaces. The well posedness is obtained in these spaces. The results obtained are related to and generalize earlier works of Rendall [35] for the Euler-Einstein system under the restriction of time symmetry and of Gamblin [17] for the simpler Euler–Poisson system.