Global dynamics for the 3D Maxwell-Dirac system

The aim of these notes is to provide an overview of the ideas in the recent proof of global well-posedness for the massive Maxwell-Dirac system in the Lorenz gauge in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> , for small and decaying initial data of limiting regularity. The result also includes an in-depth study of the asymptotic dynamics of the global solutions, which can be described as modified scattering. While heuristically we exploit the close connection between the massive Maxwell-Dirac and the wave-Klein-Gordon equations, for the proof of the results we develop 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 is derived via the method of testing with wave packets of Ifrim-Tataru.