The modified scattering for Dirac equations of scattering-critical nonlinearity

Abstract

In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long-range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Masmoudi-Shatah ([14, 15, 16]),as well as the spinorial null-structure. On the way, we recognize a long-range interaction which is responsible for a logarithmic phase correction in the modified scattering statement. This result was obtained by Cloos in his dissertation [9],via a different technique (see Remark 1.2).