# Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

## TL;DR

This paper proves global well-posedness and scattering for the energy-critical, defocusing Hartree equation with radial data in dimensions five and higher, introducing a novel Morawetz identity approach to prevent energy concentration.

## Contribution

It introduces a new method using a localized Morawetz identity to establish global results for the energy-critical Hartree equation, bypassing classical estimates.

## Key findings

- Proves global well-posedness for radial data in all dimensions n≥5.
- Establishes scattering results in the energy space.
- Develops a new Morawetz identity technique to control energy concentration.

## Abstract

We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}\Delta \Big(\frac{1}{|x|}\Big)dxdt$ in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.