Long-range scattering for 2D Dirac-Hartree equations
Kiyeon Lee, Changhun Yang
Abstract
We investigate the long-time behavior of small solutions to the Dirac-Hartree equation in two spatial dimensions. This model describes the mean-field dynamics of relativistic fermions interacting through the three-dimensional Coulomb potential |x|-1, which gives rise to long-range effects in the scattering dynamics. We prove global well-posedness and long-range scattering (modified scattering) for small initial data in weighted Sobolev spaces. In this setting, long-range scattering means that, unlike linear scattering, an additional logarithmic phase correction is required to describe the precise asymptotics of nonlinear solutions. Our approach relies on the space-time resonance method, combined with special null structures inherent in the equation. Compared to the three-dimensional case CKLY2022,cloos, the novelty lies in overcoming the weaker time decay inherent to the two dimensional problem.
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