Local existence of classical solutions for the Einstein–Euler system using weighted Sobolev spaces of fractional order

Abstract

The existence of a class ]of local in time solution of the Einstein–Euler system is proven, which include static solutions. This result is the relativistic counterpart of a similar result for the Euler–Poisson system obtained by Gamblin [6]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool of the proof is the construction and use of weighted Sobolevspace of fractional order. Moreover this tools allow to improve the regularity conditions for the solutions of the constraint and evolution equation.