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My paper “Resonant decompositions and the I-method for the cubic nonlinear Schrodinger equation on “, 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
in two spatial dimensions, thus . In that paper we used the “first generation I-method” (centred around an almost conservation law for a mollified energy ) to obtain global well-posedness in for (improving on an earlier result of by Bourgain). Here we use the “second generation I-method”, in which the mollified energy 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 . (The conjectured region is ; 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 , 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.