The fast Newtonian limit for perfect fluids

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 [10] to allow for a larger class of initial data. As in [10], the proof is based on a non-local symmetric hyperbolic formulation of the Einstein–Euler equations, which contain a singular parameter $\ep=v_T/c$ 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 $\ep\searrow 0$.

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Published 18 January 2013