M ar 2 00 7 IMPROVED INTERACTION MORAWETZ INEQUALITIES FOR THE CUBIC NONLINEAR SCHRÖDINGER EQUATION ON

We prove global well-posedness for low regularity data for the L 2 − critical defocusing nonlinear Schrödinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space H s (R 2) and any s > 2 5. This improves the previous result of Fang and Grillakis where global well-posedness was established for any s ≥ 1 2. We use the I-method to take advantage of the conservation laws of the equation. The new ingredient is an interaction Morawetz estimate similar to one that has been used to obtain global well-posedness and scattering for the cubic NLS in 3d. The derivation of the estimate in our case is technical since the smoothed out version of the solution Iu introduces error terms in the interaction Morawetz inequality. A byproduct of the method is that the H s norm of the solution obeys polynomial-in-time bounds.