The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D

Abstract

We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form $f\left(-\Delta \right)$, describing a homogeneous quantum gas. Under suitable assumptions on the interaction potential and on the momentum distribution $f$, we prove that the stationary state is asymptotically stable in dimension 2. More precisely, for any initial datum which is a small perturbation of $f\left(-\Delta \right)$ in a Schatten space, the system weakly converges to the stationary state for large times.