The fast Newtonian limit for perfect fluids

Abstract

We prove the existence of a large class of dynamical solutions to the Einstein–Euler equations for which the fluid density and spatial three-velocity converge to a solution of the Poisson–Euler equations of Newtonian gravity. The results presented here generalize those of The Newtonian limit for perfect fluids to allow for a larger class of initial data. As in The Newtonian limit for perfect fluids, the proof is based on a nonlocal symmetric hyperbolic formulation of the Einstein–Euler equations, which contain a singular parameter $\epsilon = v_T /c$ with $v_T$ a characteristic speed associated to the fluid and $c$ the speed of light. Energy and dispersive estimates on weighted Sobolev spaces are the main technical tools used to analyze the solutions in the singular limit $\epsilon \searrow 0$.