Improved Interaction Morawetz Inequalities for the Cubic Nonlinear Schrödinger Equation on ℝ2

J. Colliander, Manoussos G. Grillakis, Nikolaos Tzirakis

Type: Article

Publication Date: 2007-01-01

Citations: 39

DOI: https://doi.org/10.1093/imrn/rnm090

Abstract

We prove global well-posedness for low regularity data for the L2-critical defocusing nonlinear Schrödinger equation (NLS) in 2D. More precisely, we show that a global solution exists for initial data in the Sobolev space Hs(ℝ2) and any s>25⁠. This improves the previous result of Fang and Grillakis where global well-posedness was established for any s≥12⁠. 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 by-product of the method is that the Hs norm of the solution obeys polynomial-in-time bounds.

Locations

International Mathematics Research Notices - View
arXiv (Cornell University) - View - PDF

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