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arXiv · 2308.13159

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

Liying Tao·Tengfei Zhao
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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.

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Liying Tao, Tengfei Zhao. 2023-08-25. Almost sure scattering for defocusing energy critical Hartree equation on $\R^5$. https://arxiv.org/abs/2308.13159

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