Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

TL;DR

This paper establishes global regularity and scattering for the energy-critical Maxwell-Klein-Gordon equation in four spatial dimensions using advanced concentration compactness techniques.

Contribution

It introduces a modified profile decomposition and applies a concentration compactness/rigidity framework to the Maxwell-Klein-Gordon equation in the Coulomb gauge.

Findings

Proves global regularity and scattering for the equation.

Develops a new modified profile decomposition method.

Provides a priori bounds for solutions.

Abstract

We prove global regularity, scattering and a priori bounds for the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on (1+4)-dimensional Minkowski space. The proof is based upon a modified Bahouri-Gerard profile decomposition [1] and a concentration compactness/rigidity argument by Kenig-Merle [10], following the method developed by the first author and Schlag [20] in the context of critical wave maps.