Local well-posedness for the Maxwell-Dirac system in temporal gauge

TL;DR

This paper establishes low regularity well-posedness results for the Maxwell-Dirac system in 2+1 and 3+1 dimensions under the temporal gauge, leveraging null conditions and Tao's method for Yang-Mills equations.

Contribution

It proves well-posedness for the Maxwell-Dirac system at low regularity in the temporal gauge, utilizing null structures and adapting Tao's approach.

Findings

Well-posedness established in 2+1 and 3+1 dimensions.

Utilization of null conditions to handle nonlinearities.

Application of Tao's method for the temporal gauge.

Abstract

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in n+1 dimensions in the cases $n=3$ and $n=2$ : \begin{align*} \partial^{\mu} F_{\mu \nu} & = - \langle \psi,\alpha_{\nu} \psi \rangle \\ -i \alpha^{\mu} \partial_{\mu} \psi & = A_{\mu} \alpha^{\mu} \psi \, , \end{align*} where $ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu}$ , and $\alpha^{\mu}$ are the Dirac matrices. We assume the temporal gauge $A_0=0$ and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system in this gauge.