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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 $\mathbb{R}^n$, $n\ge 3$

Myeongju Chae 1, and  Soonsik Kwon 2,

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

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