Global existence in the critical space for the Thirring and Gross-Neveu models coupled with the electromagnetic field

TL;DR

This paper establishes global well-posedness for coupled Maxwell-Dirac-Thirring-Gross-Neveu equations in one dimension with critical initial data, simplifying previous proofs and extending results to new coupled models.

Contribution

It introduces new function spaces that simplify the proof of global existence and extends well-posedness results to coupled electromagnetic and fermionic models.

Findings

Proves global well-posedness in critical space for coupled models

Simplifies existing proofs for Thirring and Gross-Neveu models

Extends local well-posedness to quadratic Dirac equations

Abstract

We prove global well-posedness for the coupled Maxwell-Dirac-Thirring-Gross-Neveu equations in one space dimension, with data for the Dirac spinor in the critical space $L^2(\R)$. In particular, we recover earlier results of Candy and Huh for the Thirring and Gross-Neveu models, respectively, without the coupling to the electromagnetic field, but the function spaces we introduce allow for a greatly simplified proof. We also apply our method to prove local well-posedness in $L^2(\R)$ for a quadratic Dirac equation, improving an earlier result of Tesfahun and the author.