Global Well-Posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

2006/02/24 by De Silva, Daniela, Pavlović, Nataša, Staffilani, Gigliola +1 · 4 citations
#35A05 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0602560

Abstract

The initial value problem for the L² critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well posed in H^s(\Bbb T^d), for s>4/9 and s>2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \citebo2. 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 H¹(\Bbb 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, \Bbb T^d_λ = \Bbb R^d/λ\Bbb Z^d, d=1,2.

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