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.