Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 665 - 686, Volume 21, Issue 3, July 2008      doi:10.3934/dcds.2008.21.665

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J. Colliander - Department of Mathematics, University of Toronto, Toronto, Ontario M5R 1P2, Canada (email)
M. Keel - Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, United States (email)
Gigliola Staffilani - Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, United States (email)
H. Takaoka - Department of Mathematics, Kobe University, Nada, Kobe, Hyogo, Japan (email)
T. Tao - UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, United States (email)

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

Keywords:  Nonlinear Schrödinger equation, well-posedness, Strichartz estimates, resonant decomposition.
Mathematics Subject Classification:  35Q55.

Received: June 2007;      Revised: January 2008;      Available Online: April 2008.