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

### Changxing Miao

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China### Guixiang Xu

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China### Lifeng Zhao

Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

## 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 $s > \mathrm{\max }(1/ 2,4(\gamma −2)/ (3\gamma −4))$. 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.

## Cite this article

Changxing Miao, Guixiang Xu, Lifeng Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$-subcritical Hartree equation in $R^{d}$. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1831–1852

DOI 10.1016/J.ANIHPC.2009.01.003