Global well-posedness for the defocusing Hartree equation with radial data in $\mathbb R^4$

TL;DR

This paper proves global well-posedness and scattering for the defocusing Hartree equation in four dimensions with radial data, using advanced harmonic analysis techniques, for subcritical regularity levels.

Contribution

It establishes the first sharp global well-posedness results for the Hartree equation with radial data in \\mathbb{R}^4 for a range of potentials, extending previous local results.

Findings

Global well-posedness for s > s_c

Scattering results for the equation

Extension of techniques to non-critical cases

Abstract

By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schr\"odinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{ \begin{array}{ll} iu_t + \Delta u &=F(u), \quad (t,x) \in \mathbb{R} \times \mathbb{R}^4 u(0) \\ &=u_0(x)\in H^s(\mathbb{R}^4), \end{array} \right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-\gamma}$, $3< \gamma<4$, with radial data in $H^{s}(\mathbb{R}^4)$ for $s>s_c:=\gamma/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.