Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$

Details Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$ Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$

2007-02-23

arxiv.org/abs/math/0702707v2

Mathematics

Analysis of PDEs

29 pages, improved result

Scientific paper

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space $H^{s}(\mathbb R)$ for any $s>{1/3}$. This improves the result in \cite{tz}, where global well-posedness was established for any $s>{4/9}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient in our proof is an interaction Morawetz estimate for the smoothed out solution $Iu$. As a byproduct of our proof we also obtain that the $H^{s}$ norm of the solution obeys polynomial-in-time bounds.

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