The modified scattering of 2 dimensional semi-relativistic Hartree equations

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.