Uniform dispersive estimates for the semi-classical Hartree equation with long-range interaction

Abstract

In this paper, we consider the Hartree equation with smooth but long-range interaction in the semi-classical regime, in three-dimensional space. We show that the density function of small-data solution decays at the optimal rate. When the semi-classical parameter $\hbar \in (0,1]$ is fixed, our result is essentially covered by the recent work by Nguyen and You [arXiv:2408.15860]; however, the novelty of this paper is the uniformity with respect to $\hbar$. Namely, both smallness condition for initial data and bounds for the solution are independent of $\hbar$. Moreover, the argument in this paper provides a new proof of the modified scattering for the long-range nonlinear Schr\"{o}dinger equation with a Hartree type nonlinearity. Our proof relies on three main ingredients. First, we prove the boundedness of finite-time wave operators modified by phase corrections. Second, we show an $L^1$--$L^\infty$ dispersive estimate for the modified propagator. Third, we give various kinds of commutator estimates for density operators. By combining them, we can apply the usual bootstrap argument to obtain the main result.