The modified scattering of two dimensional semi-relativistic Hartree equations

In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is the cubic one convolved with the Coulomb potential \(|x|^{-1}\), and it produces the long-range interaction in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to 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 by Germain et al. (Int Math Res Not 2009(3):414–432, 2008), and Pusateri (Commun Math Phys 332(3):1203–1234, 2014). Compared to the result in three dimensional case (Pusateri 2014), weaker time decay in two dimension is one of the main obstacles.