Global well-posedness of the Maxwell-Dirac system in two space dimensions

TL;DR

This paper proves the global well-posedness of the Maxwell-Dirac system in two dimensions for initial data in the charge class, utilizing null structures and charge subcriticality.

Contribution

It extends previous results from Dirac-Klein-Gordon to the full Maxwell-Dirac system in 2D, establishing global solutions in the charge class.

Findings

Proves local well-posedness in the charge class for Maxwell-Dirac in 2D.

Establishes global well-posedness using null structure and charge subcriticality.

Builds on methods from Colliander, Holmer, and Tzirakis.

Abstract

In recent work, Gr\"unrock and Pecher proved that the Dirac-Klein-Gordon system in 2d is globally well-posed in the charge class (data in $L^2$ for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell-Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in two space dimensions, and make use of the null structure of the Maxwell part.