Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension

Abstract

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 || H^, 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 d0jl +3,7°, with d0j° +8,7' =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.

Remarks on Nonlinear Dirac Equations in One Space Dimension

“…Local existence and uniqueness of solution to (1.1) in Sobolev space H s (R) can be proved in the standard way [4,5].…”

“…The Cauchy problem of (1.3) has been studied by several authors [2,4,7,13]. The global existence of solutions to Thirring equations was studied in [4] in terms of Sobolev space H s (s ≥ 1).…”

“…The global existence of solutions to Thirring equations was studied in [4] in terms of Sobolev space H s (s ≥ 1). Low regularity well-posedness was discussed in [2,7,13] showing that there exists a time T > 0 and solution…”

Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

“…In one space dimension Delgado [9] studied the Cauchy problem for the Thirring model and the Federbusch model (a 4 × 4 system of two coupled nonlinear Dirac equations), as well as the Dirac-Klein-Gordon and the Maxwell-Dirac equations, and proves global existence of H 1 solutions.…”

“…We shall show that if λ 1 = λ 2 we have global existence without the need for any smallness assumptions. This is due to a cancellation property of the nonlinearity, similar to the one used in Deldado [9]. In the case λ 1 = −λ 2 a smallness condition in L 2 is needed for global existence, in the same spirit as Glassey [11].…”