The Maxwell-Klein-Gordon equation with scattering data

TL;DR

This paper proves the existence of global solutions to the Maxwell-Klein-Gordon system that scatter to any given localized radiation field, extending previous small data results to large data using gauge invariant methods.

Contribution

It extends the small data scattering results for the MKG system to large data cases with arbitrary size and charge, employing gauge invariant vector field techniques.

Findings

Existence of global solutions for large data MKG system

Scattering to arbitrary localized radiation fields

Extension of small data results to large data regime

Abstract

It has been shown in [Yang-Yu 2019] that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space $\mathbb{R}^{1+3}$ decay like linear solutions. One hence can define the associated radiation field on the future null infinity as the limit of $(r\underline{\alpha}, r\phi)$ along the out going null geodesics. In this paper, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem for the MKG system with general large data by using gauge invariant vector field method. We in particular extend the small data result of He in \cite{MR4299134} to a class of general large data.