Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbb{R}^{1+n}$

TL;DR

This paper proves global well-posedness and scattering for the energy-critical defocusing nonlinear Hartree equation in high dimensions without assuming radial symmetry, using advanced frequency and spatial analysis techniques.

Contribution

It introduces a modified long time perturbation theory to localize frequency and spatial properties of solutions, removing the radial assumption in prior work.

Findings

Established global well-posedness and scattering for n ≥ 5

Developed a new frequency localization method

Proved spatial concentration of minimal energy blow-up solutions

Abstract

Using the same induction on energy argument in both frequency space and spatial space simultaneously as in \cite{CKSTT07}, \cite{RyV05} and \cite{Vi05}, we obtain global well-posedness and scattering of energy solutions of defocusing energy-critical nonlinear Hartree equation in $\mathbb{R}\times \mathbb{R}^n$($n\geq 5$), which removes the radial assumption on the data in \cite{MiXZ07a}. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition \ref{freqdelocaimplystbound} and Corollary \ref{frequencylocalization}) of the minimal energy blow up solutions, which can not be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that $L^{\frac{2n}{n-2}}_x$-norm of minimal energy blow up solutions is bounded from below, the $L^{\frac{2n}{n-2}}_x$-norm is larger than the potential energy.