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Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on 2 . (English) Zbl 1142.35085
Authors’ summary: 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 ( 2 ) and for any s>2 5. This improves the previous result of Y. F. Fang and M. G. Grillakis [ J. Hyperbolic Differ. Equ. 4, No. 2, 233–257 (2007; Zbl 1122.35132)] where global well-posedness was established for any s1 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 by-product of the method is that the H s norm of the solution obeys polynomial-in-time bounds.

35Q55NLS-like (nonlinear Schrödinger) equations
35B45A priori estimates for solutions of PDE
35D10Regularity of generalized solutions of PDE (MSC2000)
35P25Scattering theory (PDE)
35B45A priori estimates for solutions of PDE