Regularity results for the spherically symmetric Einstein-Vlasov system

TL;DR

This paper establishes global existence results for the spherically symmetric Einstein-Vlasov system in Schwarzschild and maximal-isotropic coordinates, extending previous bounds to non-compact data and removing certain assumptions about singularity formation.

Contribution

It introduces a new approach that improves regularity estimates, allowing for non-compact initial data and avoiding complex pointwise matter term treatments.

Findings

Global existence outside the centre in both coordinate systems.

Extended bounds on momentum support for non-compact initial data.

Proved no singularities form in Schwarzschild time under specific conditions.

Abstract

The spherically symmetric Einstein-Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the centre in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in \cite{RRS} for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in \cite{AR1}. In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as $3m\leq r.$ This removes an additional assumption made in \cite{A1}. Our result in maximal-isotropic coordinates is analogous to the result in \cite{R1}, but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.