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

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