Global well-posedness for the L^2-critical Hartree equation on \mathbb{R}^n, n\ge 3

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November 2009, 8(6): 1725-1743. doi: 10.3934/cpaa.2009.8.1725

Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$

1.	Department of Applied Mathematics, Hankyong National University, Ansong 456-749, South Korea
2.	Department of Mathematics, Princeton University, Princeton, NJ 08545, United States

Received October 2008 Revised May 2009 Published August 2009

Abstract
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We consider the initial value problem for the $L^2$-critical defocusing Hartree equation in $\mathbb{R}^n$, $n\ge 3$. We show that the problem is globally well posed in $H^s(\mathbb{R}^n)$ when $ 1 > s > \frac{2(n-2)}{3n-4}$. We use the "I-method" following [9] combined with a local in time Morawetz estimate for the smoothed out solution $I\phi$ as in [7].

Keywords: I-method, Global well-posedness, almost conservation law., Hartree equation.

Mathematics Subject Classification: 35Q5.

Citation: Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725

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