Asymptotic properties of solutions of the Maxwell Klein Gordon equation with small data

TL;DR

This paper establishes decay estimates for small solutions to the Maxwell Klein-Gordon equations with non-zero charge in four-dimensional spacetime, providing a simpler proof that avoids complex estimates used previously.

Contribution

It offers a simplified proof of peeling estimates for the Maxwell Klein-Gordon equations with small data, removing the need for fractional Morawetz estimates and employing a gauge-invariant approach.

Findings

Achieved decay rates matching previous results by Lindblad and Sterbenz.

Simplified the proof by eliminating fractional Morawetz estimates.

Extended results to cases with compactly supported scalar fields.

Abstract

We prove peeling estimates for the small data solutions of the Maxwell Klein Gordon equations with non-zero charge and with a non-compactly supported scalar field, in $(3+1)$ dimensions. We obtain the same decay rates as in an earlier work by Lindblad and Sterbenz, but giving a simpler proof. In particular we dispense with the fractional Morawetz estimates for the electromagnetic field, as well as certain space-time estimates. In the case that the scalar field is compactly supported we can avoid fractional Morawetz estimates for the scalar field as well. All of our estimates are carried out using the double null foliation and in a gauge invariant manner.