Modified scattering for the three dimensional Maxwell-Dirac system

TL;DR

This paper establishes global well-posedness and modified scattering for the 3D Maxwell-Dirac system with small initial data, using a novel approach linked to wave-Klein-Gordon equations and wave packet testing.

Contribution

It introduces a new method at the Dirac equation level to prove modified scattering, connecting Maxwell-Dirac with wave-Klein-Gordon dynamics.

Findings

Proves global well-posedness for small initial data.

Derives a precise asymptotic description of solutions inside the light cone.

Develops a novel approach using wave packet testing for Dirac equations.

Abstract

In this work we prove global well-posedness for the massive Maxwell-Dirac system in the Lorenz gauge in $\mathbb{R}^{1+3}$, for small, sufficiently smooth and decaying initial data, as well as modified scattering for the solutions. Heuristically we exploit the close connection between the massive Maxwell-Dirac and the wave-Klein-Gordon equations, while developing a novel approach which applies directly at the level of the Dirac equations. The modified scattering result follows from a precise description of the asymptotic behavior of the solutions inside the light cone, which we derive via the method of testing with wave packets of Ifrim-Tataru.