Modified Scattering for the Three Dimensional Maxwell-Dirac System

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.