Global finite energy solutions of the Maxwell-scalar field system on the Einstein cylinder

TL;DR

This paper establishes the global existence and uniqueness of finite energy solutions for the Maxwell-scalar field system on the Einstein cylinder, utilizing conformal methods, null form estimates, and localization techniques.

Contribution

It introduces a novel combination of conformal patching, localization, and null form estimates to prove global solutions in Lorenz gauge on the Einstein cylinder.

Findings

Proves global finite energy solutions exist and are unique.

Demonstrates regularity preservation of energy-carrying components.

Identifies small regularity losses due to gauge and foliation issues.

Abstract

We prove the existence and uniqueness of global finite energy solutions of the Maxwell-scalar field system in Lorenz gauge on the Einstein cylinder. Our method is a combination of a conformal patching argument, the finite energy existence theorem in Lorenz gauge on Minkowski space of Selberg and Tesfahun, a careful localization of finite energy data, and null form estimates of Foschi-Klainerman type. Although we prove that the energy-carrying components of the solution maintain regularity, due to the incompleteness of the null structure in Lorenz gauge and the nature of our foliation-change arguments we find small losses of regularity in both the scalar field and the potential.