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 Hs(double-struck T signd), for s > 4/9 and s > 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in . 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(double- struck T signd) 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, double-struck T sign λd = ℝd/λℤd, d = 1, 2.
- Global well-posedness
- Nonlinear Schrödinger equation
- Nonlinear dispersive equations
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics