# Global well-posedness and scattering for the defocusing $H^{frac12}$-subcritical Hartree equation in $mathbb{R}^d$ - Mathematics > Analysis of PDEs

Global well-posedness and scattering for the defocusing $H^{frac12}$-subcritical Hartree equation in $mathbb{R}^d$ - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: We prove the global well-posedness and scattering for the defocusing$H^{\frac12}$-subcritical that is, $2<\gamma<3$ Hartree equation with lowregularity data in $\mathbb{R}^d$, $d\geq 3$. Precisely, we show that a uniqueand global solution exists for initial data in the Sobolev space$H^s\big\mathbb{R}^d\big$ with $s>4\gamma-2-3\gamma-4$, which alsoscatters in both time directions. This improves the result in \cite{ChHKY},where the global well-posedness was established for any$s>\max\big1-2,4\gamma-2-3\gamma-4\big$. The new ingredients in our proofare that we make use of an interaction Morawetz estimate for the smoothed outsolution $Iu$, instead of an interaction Morawetz estimate for the solution$u$, and that we make careful analysis of the monotonicity property of themultiplier $m\xi\cdot < \xi>^p$. As a byproduct of our proof, we obtain thatthe $H^s$ norm of the solution obeys the uniform-in-time bounds.

Author: ** Changxing Miao, Guixiang Xu, Lifeng Zhao**

Source: https://arxiv.org/