Global well-posedness of high dimensional Maxwell-Dirac for small critical data

TL;DR

This paper establishes the global well-posedness and modified scattering for the massless Maxwell-Dirac equation in high dimensions with small critical data, by revealing null structures and solving the covariant Dirac equation.

Contribution

It introduces a novel approach by exploiting the analogy between Maxwell-Dirac and Maxwell-Klein-Gordon to prove well-posedness in high dimensions.

Findings

Proved global well-posedness for small critical data in high dimensions.

Demonstrated modified scattering behavior of solutions.

Uncovered null structure in Maxwell-Dirac in Coulomb gauge.

Abstract

In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.