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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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