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

Daniela De Silva; Nataša Pavlović; Gigliola Staffilani; Nikolaos Tzirakis

doi:10.3934/dcds.2007.19.37

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2007, Volume 19, Issue 1: 37-65. Doi: 10.3934/dcds.2007.19.37

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

1.
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218
2.
Department of Mathematics, Princeton University, Princeton, NJ 08544-1000
3.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307
4.
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4

Received: April 30, 2006

Revised: March 31, 2007

Published: December 2006

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Abstract

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, $T^{d}_\lambda = R^{d}/{\lambda Z^{d}}$, $d=1,2$.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B30.

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

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