2007/03/20 by J. Colliander, Manoussos G. Grillakis, M. Grillakis +5 · 3 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.math/0703606
21 pages
arxiv created 2007/03/20 · openalex publication_date 2007/03/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(\mathbb R2) 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 Hs norm of the solution obeys polynomial-in-time bounds.