Rough solutions of Einstein vacuum equations in CMCSH gauge

TL;DR

This paper establishes local well-posedness for very rough solutions to Einstein vacuum equations in CMC gauge, using a novel approach that avoids paradifferential regularization and employs commuting vector fields for Strichartz estimates.

Contribution

It introduces a new method to prove Strichartz estimates directly for rough Einstein metrics without paradifferential regularization.

Findings

Proves local well-posedness for solutions in $H^s$, s>2.

Develops a direct approach for geometric wave equations on rough backgrounds.

Avoids standard paradifferential regularization techniques.

Abstract

In this paper, we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in $H^s, s>2$. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $\bg$, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation $\Box_\bg \phi=0$ directly.