Uniform Energy Bound for Maxwell-Higgs Equations on Reissner-Nordstr\"om Spacetimes

TL;DR

This paper establishes uniform energy bounds for the Maxwell-Higgs equations in the exterior of Reissner-Nordström black holes, combining decay estimates, Sobolev embeddings, and inequalities to control the fields.

Contribution

It introduces a novel approach to proving uniform energy bounds for the Maxwell-Higgs system on Reissner-Nordström backgrounds using integrated local energy decay and Sobolev techniques.

Findings

Derived $L^\infty$ bounds for the fields.

Established bounds for the conformal energy of the system.

Demonstrated decay estimates in the exterior region of black holes.

Abstract

In this paper, we prove the uniform energy bound for the Maxwell-Higgs system in the exterior region of Reissner-Nordstr\"om black holes. By employing an integrated local energy decay (ILED) estimate in combination with the Sobolev embedding theorem on a compact Riemannian manifold, we derived $L^\infty$ bounds for the fields. These results were then used to obtain a bound for the conformal energy of the system using the Cauchy-Schwarz inequality and Hardy-type inequalities.