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

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 ${\mathbb{R}}^d$, $d⩾3$. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space $H^s({\mathbb{R}}^d)$ with $s>4(\gamma -2)/(3\gamma -4)$, which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], 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.