Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\R^2$
| dc.creator | Colliander, J. | |
| dc.creator | Grillakis, M. | |
| dc.creator | Tzirakis, N. | |
| dc.date | 2007-03-20 | |
| dc.date.accessioned | 2026-07-07T07:52:54Z | |
| dc.date.available | 2026-07-07T07:52:54Z | |
| dc.description | 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 $s \geq {1/2}$. 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. | |
| dc.description | 21 pages | |
| dc.identifier | https://arxiv.org/abs/math/0703606 | |
| dc.identifier | http://arxiv.org/abs/math/0703606 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/126050 | |
| dc.subject | Analysis of PDEs | |
| dc.title | Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\R^2$ | |
| dc.type | text |