The modified scattering for Dirac equations of scattering-critical nonlinearity

TL;DR

This paper proves global well-posedness and modified scattering for small solutions of a 3D Maxwell-Dirac system with long-range Hartree nonlinearity, using weighted energy estimates and resonance analysis.

Contribution

It establishes the first modified scattering result for the 3D Maxwell-Dirac system with long-range nonlinearity under the Lorenz gauge.

Findings

Global well-posedness for small solutions.

Modified scattering with logarithmic phase correction.

Identification of long-range interaction effects.

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, 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.