The modified scattering of 2 dimensional semi-relativistic Hartree equations

TL;DR

This paper studies the long-term behavior of small solutions to 2D semi-relativistic Hartree equations with Coulomb potential, demonstrating global well-posedness and modified scattering despite weaker decay in two dimensions.

Contribution

It establishes the global well-posedness and modified scattering for small solutions in 2D semi-relativistic Hartree equations, extending understanding of long-range interactions in lower dimensions.

Findings

Proves global existence of small solutions.

Shows solutions exhibit modified scattering behavior.

Addresses challenges due to weaker decay in 2D.

Abstract

In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the Coulomb potential 1/|x|, and it produces the long-range interaction in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to a modified scattering states, although they decay as linear solutions. We show the global well-posedness and the modified scattering for small solutions in weighted Sobolev spaces. Our proof follows a road map of exploiting the space-time resonance developed by Germain, Masmoudi, and Shatah. Compared to the result in three dimensional case by Pusateri, weaker time decay in two dimension is one of the main obstacles.