Global well-posedness for the massive Maxwell-Klein-Gordon equation with small critical Sobolev data

TL;DR

This paper establishes global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in higher dimensions with small critical Sobolev data, extending previous massless results to the massive case.

Contribution

It generalizes the global parametrix construction and functional framework to the massive Klein-Gordon setting, handling trilinear cancellations and null form bounds.

Findings

Proved global well-posedness for small data in higher dimensions.

Extended massless case results to the massive case with m^2 > 0.

Developed sharp null form bounds and handled logarithmic divergences.

Abstract

In this paper we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small critical Sobolev norm. This extends to the general case $ m^2 > 0 $ the results of Krieger-Sterbenz-Tataru ($d=4,5 $) and Rodnianski-Tao ($ d \geq 6 $), who considered the case $ m=0$. We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon-Sterbenz. To treat it one needs sharp $ L^2 $ null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru-Herr. To overcome logarithmic divergences we rely on an embedding property of $ \Box^{-1} $ in conjunction with endpoint Strichartz estimates in Lorentz spaces.