Global solution for Massive Maxwell-Klein-Gordon equations with large Maxwell field

TL;DR

This paper establishes the global behavior of the massive Maxwell-Klein-Gordon system with large initial data, revealing a hidden cancellation in the equations that allows for sharp control of the fields.

Contribution

It introduces a gauge-independent method to analyze the system with large Maxwell fields, uncovering a cancellation that improves control over the solutions.

Findings

Global existence for large initial data established

Hidden cancellation in Maxwell equations identified

Sharp control of the Maxwell field achieved

Abstract

We derive the global dynamic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field) with a general, unrestrictive class of data, in particular, for Maxwell field of arbitrary size, and by a gauge independent method. Due to the critical slow decay expected for the Maxwell field, the scalar field exhibits a loss of decay at the causal infinities within an outgoing null cone. To overcome the difficulty caused by such loss in the energy propagation, we uncover a hidden cancellation contributed by the Maxwell equation, which enables us to obtain the sharp control of the Maxwell field under a rather low regularity assumption on data. Our method can be applied to other physical field equations, such as the Einstein equations for which a similar cancellation structure can be observed.