Phase mixing estimates for the nonlinear Hartree equation of infinite rank

In this paper, we prove the phase mixing estimates for the density and its derivatives associated with the nonlinear Hartree equation around certain translation-invariant equilibria. Given a defocusing short-range interaction potential, we provide a precise criterion for the Penrose–Lindhard stability based on the marginal of the equilibrium. For linearly stable equilibria, pointwise decay estimates of the Green function associated with the linearized operator in Fourier space are established. The proof of phase mixing estimates is obtained through a nonlinear iterative scheme. An alternative proof of scattering is also provided.