Phase mixing estimates for the nonlinear Hartree equation of infinite rank

TL;DR

This paper establishes phase mixing estimates and decay properties for the nonlinear Hartree equation near stable equilibria, using a nonlinear iterative scheme and providing criteria for stability.

Contribution

It introduces a new criterion for Penrose--Lindhard stability and offers a novel proof of scattering for the nonlinear Hartree equation.

Findings

Derived phase mixing estimates for density and derivatives.

Provided a stability criterion based on the equilibrium's marginal.

Established pointwise decay estimates for the linearized operator.

Abstract

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.