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).
Citation
Yonggeun Cho. Soonsik Kwon. Kiyeon Lee. Changhun Yang. "The modified scattering for Dirac equations of scattering-critical nonlinearity." Adv. Differential Equations 29 (3/4) 179 - 222, March/April 2024. https://doi.org/10.57262/ade029-0304-179
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