Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$

J. Colliander; M. Keel; Gigliola Staffilani; H. Takaoka; T. Tao

doi:10.3934/dcds.2008.21.665

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2008, Volume 21, Issue 3: 665-686. Doi: 10.3934/dcds.2008.21.665 open access

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Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$

1.
Department of Mathematics, University of Toronto, Toronto, Ontario M5R 1P2
2.
Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455
3.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307
4.
Department of Mathematics, Kobe University, Nada, Kobe, Hyogo
5.
UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555

Received: May 31, 2007

Revised: December 31, 2007

Published: August 2008

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Abstract

The initial value problem for the cubic defocusing nonlinear Schrödinger equation $i \partial_t u + \Delta u = |u|^2 u$ on theplane is shown to be globally well-posed for initial data in $H^s (\mathbb{R}^2)$ provided $s>1/2$. The same result holds true for theanalogous focusing problem provided the mass of the initial data issmaller than the mass of the ground state. The proof relies upon analmost conserved quantity constructed using multilinear correctionterms. The main new difficulty is to control the contribution ofresonant interactions to these correction terms. The resonantinteractions are significant due to the multidimensional setting ofthe problem and some orthogonality issues which arise.

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Mathematics Subject Classification: 35Q55.

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Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$

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