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## Resonant decompositions and the I-method for the cubic nonlinear Schrodinger equation on R^2

24 April, 2007 in math.AP, paper | Tags: dispersive equations, I-method, NLS, Schrodinger equation | by Terence Tao | 2 comments

My paper “Resonant decompositions and the I-method for the cubic nonlinear Schrodinger equation on ${\Bbb R}^2$“, with Jim Colliander, Mark Keel, Gigliola Staffilani, and Hideo Takaoka (aka the “I-team“), has just been uploaded to the arXiv, and submitted to DCDS-A. In this (long-delayed!) paper, we improve our previous result on the global well-posedness of the cubic non-linear defocusing Schrödinger equation

$i u_t+ \Delta u = |u|^2 u$

in two spatial dimensions, thus $u: {\Bbb R} \times {\Bbb R}^2 \to {\Bbb C}$. In that paper we used the “first generation I-method” (centred around an almost conservation law for a mollified energy $E(Iu)$) to obtain global well-posedness in $H^s({\Bbb R}^2)$ for $s > 4/7$ (improving on an earlier result of $s > 2/3$ by Bourgain). Here we use the “second generation I-method”, in which the mollified energy $E(Iu)$ is adjusted by a correction term to damp out “non-resonant interactions” and thus lead to an improved almost conservation law, and ultimately to an improvement of the well-posedness range to $s > 1/2$. (The conjectured region is $s \geq 0$; beyond that, the solution becomes unstable and even local well-posedness is not known.) A similar result (but using Morawetz estimates instead of correction terms) has recently been established by Colliander-Grillakis-Tzirakis; this attains the superior range of $s > 2/5$, but in the focusing case it does not give global existence all the way up to the ground state due to a slight inefficiency in the Morawetz estimate approach. Our method is in fact rather robust and indicates that the “first-generation” I-method can be pushed further for a large class of dispersive PDE.

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