Global well-posedness for the $L^2$ critical Hartree equation on $\bbr^n$, $n\ge 3$

TL;DR

This paper proves global well-posedness for the L^2-critical defocusing Hartree equation in higher dimensions using the I-method and Morawetz estimates, extending the understanding of solution behavior in Sobolev spaces.

Contribution

It establishes global well-posedness for the Hartree equation in H^s for a specific range of s, employing a novel combination of the I-method and Morawetz estimates.

Findings

Global well-posedness in H^s for 1>s>2(n-2)/(3n-4)

Application of the I-method to the Hartree equation

Use of local Morawetz estimates for smoothing

Abstract

We consider the initial value problem for the L^2-critical defocusing Hartree equation in R^n, n \ge 3. We show that the problem is globally well posed in H^s(R^n) when 1>s> \frac{2(n-2)}{3n-4}$. We use the "I-method" combined with a local in time Morawetz estimate for the smoothed out solution.