University of Toronto
               PDE/Applied Math/Analysis Seminar
                 Monday November 7, 3:10-4pm
                       1230 Bahen Center      




SPEAKER:  Nikos Tzirakis

TITLE:    Global Well-Posedness for the L^2-critical periodic nonlinear 
Schrodinger equation in 1d and 2d.
          

ABSTRACT: In this talk I will consider the initial value problem
for the $L^{2}$-critical semilinear Schr\"odinger equation, with 
periodic boundary conditions. I will show that the problem is globally
well-posed in $H^{s}({\Bbb T^{d}})$, for any $s>4/9$ and any $s>4/7$ in 1d
and 2d respectively, matching the results that are already known for
initial data on $\Bbb R$ and $\Bbb R^{2}$. The periodic problem is well 
studied and the previous best known global well-posedness results were due
to J. Bourgain. I will use the "I-method". This method allows 
one to define a modification of the energy functional that is
well defined for initial data below the $H^{1}({\Bbb T^{d}})$ threshold.
Apart from that, the main new ingredient is the proof of new bilinear
refinements of Strichartz estimates that hold true for periodic solutions 
defined on $[0, \lambda]$, $\lambda>0$. This is joint work with
D. De Silva, N. Pavlovic and G. Staffilani.