Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

TL;DR

This paper reveals the hidden null structure of the Maxwell-Dirac system in Lorenz gauge and uses it to establish almost optimal local well-posedness results at critical regularity levels.

Contribution

It uncovers the full null structure of the Maxwell-Dirac system and proves frequency-localized estimates leading to near-optimal well-posedness results.

Findings

Identified the null structure involving tri- and quadrilinear forms.

Established frequency-localized $L^2$ estimates at scale-invariant regularity.

Achieved almost optimal local well-posedness of the Maxwell-Dirac system.

Abstract

We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri- and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequency-localized $L^2$ space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local well-posedness of the system by iteration.