Global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon system in the Lorenz gauge

TL;DR

This paper proves global well-posedness and scattering for the energy-critical Maxwell-Klein-Gordon system in the Lorenz gauge by introducing angular regularity, achieving results at the critical regularity level.

Contribution

It is the first to establish global existence and scattering for (MKG) in the Lorenz gauge at the scaling critical regularity.

Findings

Global well-posedness for small data in critical space

Scattering results for solutions in the Lorenz gauge

Overcoming nonlinearity without null structure using angular regularity

Abstract

We study initial value problem of the $(1+4)$-dimensional Maxwell-Klein-Gordon system (MKG) in the Lorenz gauge. Since (MKG) in the Lorenz gauge does not possess an obvious null structure, it is not easy to handle the nonlinearity. To overcome this obstacle, we impose an additional angular regularity. In this paper, we prove global well-posedness and scattering of (MKG) for small data in a scale-invariant space which has extra weighted regularity in the angular variables. Our main improvement is to attain the scaling critical regularity exponent and prove global existence of solutions to (MKG) in the Lorenz gauge.