Local well-posedness for low regularity data for the higher-dimensional Maxwell-Klein-Gordon system in Lorenz gauge

TL;DR

This paper proves local well-posedness for the Maxwell-Klein-Gordon system with low regularity data in higher dimensions, leveraging null structures and advanced wave-Sobolev space estimates.

Contribution

It extends local well-posedness results to higher dimensions and low regularity data using null structures and Klainerman-Selberg solution spaces.

Findings

Null structure exists in Lorenz gauge for the system

Local well-posedness established for low regularity data in dimensions n ≥ 4

Utilizes wave-Sobolev space estimates and Klainerman-Selberg spaces

Abstract

The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in $n$ space dimensions ($n \ge 4$) is shown to be locally well-posed for low regularity (large) data. The result relies on the null structure for the main bilinear terms which was shown to be not only present in Coulomb gauge but also in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces. Crucial for the improvement are the solution spaces introduced by Klainerman-Selberg. Preliminary results were already contained in arXiv:1705.00599.