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

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

2007-03-20

arxiv.org/abs/math/0703606v1

Mathematics

Analysis of PDEs

21 pages

Scientific paper

We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schr\"odinger 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 $s \geq {1/2}$. 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.

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