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Zbl 1142.35085
Colliander, James; Grillakis, Manoussos; Tzirakis, Nikolaos
Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\mathbb R^{2}$. (English)
[J] Int. Math. Res. Not. 2007, No. 23, Article ID rnm090, 30 p. (2007). ISSN 1073-7928; ISSN 1687-0247/e

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 $H^s(\mathbb R^{2})$ and for any $s> \frac 2 5$. This improves the previous result of {\it Y. F. Fang} and {\it M. G. Grillakis} [ J. Hyperbolic Differ. Equ. 4, No. 2, 233--257 (2007; Zbl 1122.35132)] where global well-posedness was established for any $s \geq \frac 1 2$. 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 by-product of the method is that the $H^s$ norm of the solution obeys polynomial-in-time bounds.
[A. D. Osborne (Keele)]

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
35B45 A priori estimates
35D10 Regularity of generalized solutions of PDE
35P25 Scattering theory (PDE)
35B45 A priori estimates

Keywords: nonlinear Schrödinger equation; global well-posedness; Morawetz estimate; interaction Morawetz inequality; Strichartz estimate

Citations: Zbl 1122.35132

Cited in: Zbl 1178.35344 Zbl 1185.35249

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