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

Sigmund Selberg

doi:10.3934/dcds.2018107

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2018, Volume 38, Issue 5: 2555-2569. Doi: 10.3934/dcds.2018107

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

Sigmund Selberg

Department of Mathematics, University of Bergen, PO Box 7803, 5020 Bergen, Norway

Received: June 30, 2017

Published: June 2018

The author thanks Jean-Christophe Merle for his hospitality during the author's visit to the University of Vechta, where the main part of the research reported here was carried out. The author also thanks the referee for many useful comments and suggestions

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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 $L^2(\mathbb{R})$. 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 $L^2(\mathbb{R})$ for a quadratic Dirac equation, improving an earlier result of Tesfahun and the author.

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Mathematics Subject Classification: Primary: 35Q40, 35L60, 35L70.

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