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

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 $H^s({\mathbb{T}}^d)$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. 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 $H^1({\mathbb{T}}^d)$ 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, ${\mathbb{T}}_{\lambda }^d={\mathbb{R}}^d/\lambda {\mathbb{Z}}^d$, $d=1,2$.

28 pages, 3 figures