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March  2007, 19(1): 37-65. doi: 10.3934/dcds.2007.19.37

Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

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

1. 

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United States
2. 

Department of Mathematics, Princeton University, Princeton, NJ 08544-1000
3. 

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

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4, Canada

Received  May 2006 Revised  April 2007 Published  June 2007

The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well-posed in $H^{s}( T^{d} )$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [4]. We use the "$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}(T^{d} )$ threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $T^{d}_\lambda = R^{d}/{\lambda Z^{d}}$, $d=1,2$.
Keywords: Global well-posedness, nonlinear Schrödinger equation., nonlinear dispersive equations.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B3.
Citation: Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37
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