Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^2$

Abstract

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 $H^s({\mathbb{R}}^2)$ and any $s>2/5$. This improves the previous result of Fang and Grillakis where global well-posedness was established for any $s\ge 1/2$. 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.

21 pages

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