# Sharp ill-posedness for the Maxwell-Dirac equations in one space dimension

## TL;DR

This paper establishes the precise regularity threshold for well-posedness of the Maxwell-Dirac equations in one dimension, showing well-posedness in the charge class and ill-posedness below it, with explicit counterexamples.

## Contribution

It proves sharp well-posedness results for the Maxwell-Dirac equations in one dimension, including explicit ill-posedness for lower regularity data, without relying on null structure.

## Key findings

- Well-posedness in the charge class ($L^2$ data)
- Ill-posedness for data in $H^s$ with $s<0$
- Ill-posedness for data in $L^p$ with $1 \,< p < 2$

## Abstract

The Maxwell-Dirac equations in one space dimension are proved to be well posed in the charge class, that is, with $L^2$ data for the spinor. We also prove that this result is sharp, in the sense that well-posedness fails for spinor data in $H^s$ with $s<0$, as well as in $L^p$ with $1 \le p < 2$. More precisely, we give an explicit example of such data for which no local solution can exist. Our proof of well-posedness applies to a class of systems which includes also the Dirac-Klein-Gordon system, but it does not require any null structure in the system.