American Institute of Mathematical Sciences

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

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