Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

TL;DR

This paper proves that the energy-critical Maxwell-Klein-Gordon equation in 4+1 dimensions is globally well-posed and solutions scatter for all finite energy initial data, completing a three-part research series.

Contribution

It establishes the nonexistence of nontrivial stationary or self-similar solutions, thereby confirming global well-posedness and scattering for the MKG equation in 4+1 dimensions.

Findings

Global well-posedness and scattering are proven for all finite energy data.

Nonexistence of nontrivial stationary or self-similar solutions is shown.

The proof completes a three-paper sequence on this topic.

Abstract

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.