Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$-subcritical Hartree equation in ${\mathbb{R}}^d$

Abstract

We prove the global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$-subcritical (that is, $2<\gamma <3$) Hartree equation with low regularity data in ${\mathbb{R}}^d$, $d\ge 3$. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space $H^s\left({\mathbb{R}}^d\right)$ with $s>4(\gamma -2)/(3\gamma -4)$, which also scatters in both time directions. This improves the result in \cite{ChHKY}, where the global well-posedness was established for any $s>\max (1/2,4(\gamma -2)/(3\gamma -4))$. The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution $Iu$, instead of an interaction Morawetz estimate for the solution $u$, and that we make careful analysis of the monotonicity property of the multiplier $m(\xi )\cdot <\xi {>}^p$. As a byproduct of our proof, we obtain that the $H^s$ norm of the solution obeys the uniform-in-time bounds.

arXiv comment24 pages,1 figure

References in corpus (3)

• Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
• Global well-posedness and scattering for the mass-critical Hartree equation with radial data
• Long Range Scattering and Modified Wave Operators for some Hartree Type Equations II

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