Global Well-Posedness for a periodic nonlinear Schr\"odinger equation in 1D and 2D

The initial value problem for the $L^2$ critical semilinear Schr"odinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^s({\mathbb{T}}^d)$, for $s>4\mathrm{/}9$ and $s>2\mathrm{/}3$ 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 $H^{1}({\Bbb T^{d}})$ 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, ${\mathbb{T}}_{\lambda }^d={\mathbb{R}}^d\mathrm{/}\lambda {\mathbb{Z}}^d$, $d=1,2$.