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

Arthur Touati

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arXiv · 2101.09093

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

Arthur Touati

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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.

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Arthur Touati. 2021-01-22. Einstein vacuum equations with $\mathbb{U}(1)$ symmetry in an elliptic gauge : local well-posedness and blow-up criterium. https://arxiv.org/abs/2101.09093

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