Einstein vacuum equations with $\mathbb{U}(1)$ symmetry in an elliptic gauge : local well-posedness and blow-up criterium

TL;DR

This paper establishes local well-posedness and a blow-up criterion for Einstein vacuum equations with $(1)$ symmetry in an elliptic gauge, facilitating the analysis of high-frequency solutions.

Contribution

It introduces a framework in an elliptic gauge for Einstein vacuum equations with symmetry, enabling the study of high-frequency solutions and their potential blow-up.

Findings

Existence of unique local solutions at the $H^3$ level.

Blow-up criterion established at the $H^2$ level.

Framework suited for high-frequency solution analysis.

Abstract

In this article, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying $\mathbb{U}(1)$ symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the $H^3$ level. In addition, we prove a blow-up criterium at the $H^2$ level. Our main objective is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.