Almost sure scattering for defocusing energy critical Hartree equation on $\R^5$

TL;DR

This paper proves that solutions to the defocusing energy-critical Hartree equation in five dimensions scatter almost surely for initial data in any Sobolev space, using advanced probabilistic and analytical techniques.

Contribution

It introduces an alternative proof for the interaction Morawetz estimate and extends almost sure scattering results to the nonlocal Hartree equation in 5D.

Findings

Almost sure scattering for initial data in any Sobolev space

New proof of interaction Morawetz estimate for nonlocal nonlinearity

Extension of probabilistic scattering techniques to Hartree equation

Abstract

We consider the defocusing energy-critical Hartree equation $i\pa_tu+\Delta u=(|\cdot|^{-4}\ast|u|^2)u$ in spatial dimension $d=5$ and prove almost sure scattering with initial data $u_0\in H^s_x(\R^5)$ for any $s\in\R$. The proof relies on the modified interaction Morawetz estimate, the stability theories, the ``Narrowed'' Wiener randomization. We are inspired to consider this problem by the work of Shen-Soffer-Wu \cite{Shen-Soffer-Wu 1}, which treated the analogous problem for the energy-critical Schr\"{o}dinger equation. The new ingredient in this paper are that we take an alternative proof to give the interaction Morawetz estimate. And the nonlocal nonlinearity term will bring some difficulties.