Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\R^2$
J. Colliander - Department of Mathematics, University of Toronto, Toronto, Ontario M5R 1P2, Canada (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.
Received: June 2007; Revised: January 2008; Published: April 2008.
2012 Impact Factor1.005