Global Solutions of the Cauchy Problem for the (Classical) Coupled Maxwell-Dirac and Other Nonlinear Dirac Equations in One Space Dimension

Abstract

The existence of global solutions is proved for the Maxwell-Dirac equations, for the Thirring model (Dirac equation with vector self-interaction), for the Klein-Gordon-Dirac equations and for two Dirac equations coupled through a vector-vector interaction (Federbusch model) in one space dimension. The proof is based on charge conservation, and depends on an “a priori” estimate of ${\left \| {} \right \|_\infty }$ for the Dirac field. This estimate is obtained only on the basis of algebraical properties of the nonlinear term, and allows us to simplify the proofs of global existence. We obtain it by computing ${\partial _0}{j^1} + {\partial _1}{j^0}$, with ${\partial _0}{j^0} + {\partial _1}{j^1} = 0$ being the continuity equation which expresses charge conservation. We also prove global existence for the Thirring and Federbusch models coupled in standard form with the electromagnetic field.