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

Abstract

We prove the existence of a class of local in time solutions, including static solutions, of the Einstein–Euler system. This result is the relativistic generalisation of a similar result for the Euler–Poisson system obtained by Gamblin (1993). 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 in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article. To cite this article: U. Brauer, L. Karp, C. R. Acad. Sci. Paris, Ser. I 345 (2007).