Local Existence of Solutions of Self Gravitating Relativistic Perfect Fluids

TL;DR

This paper establishes local existence, uniqueness, and well-posedness for solutions to the Einstein--Euler system describing self-gravitating perfect fluids in asymptotically flat spacetimes, addressing technical challenges from fractional power equations.

Contribution

It introduces a novel approach using weighted Sobolev spaces of fractional order to handle the system's technical difficulties and extends classical regularity results to this setting.

Findings

Proves local well-posedness for Einstein--Euler system with fractional power state equation.

Handles non-smooth zero order terms with weighted Sobolev spaces.

Achieves regularity bounds similar to classical vacuum Einstein equations.

Abstract

This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein--Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity, and that the pressure is a fractional power of the energy density. In this setting we prove a local in time existence, uniqueness and well-posedness of classical solutions. The zero order term of our system contains an expression which might not be a $C^\infty$ function and therefore causes an additional technical difficulty. In order to achieve our goals we use a certain type of weighted Sobolev space of fractional order. Previously the authors constructed an initial data set for these of systems in the same type of weighted Sobolev spaces. We obtain the same lower bound for the regularity as the one of the classical result of Hughes, Kato and Marsden for the vacuum Einstein equations. However, due to the presence of an equation of state with fractional power, the regularity is bounded from above.