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December  2007, 6(4): 1023-1041. doi: 10.3934/cpaa.2007.6.1023

Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions

Daniela De Silva 1, , Nataša Pavlović 2, , Gigliola Staffilani 3, and  Nikolaos Tzirakis 4,

1. 

Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027, United States
2. 

Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712, United States
3. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307
4. 

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, United States

Received  October 2006 Revised  June 2007 Published  September 2007

The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\mathbb R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^s(\mathbb R^n )$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the "$I$-method" combined with a local in time Morawetz estimate.
Keywords: global well-posedness, Schrödinger equations, $I$-method, Morawetz estimates..
Mathematics Subject Classification: 35Q5.
Citation: Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023
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