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

Daniela De Silva; Nataša Pavlović; Gigliola Staffilani; Nikolaos Tzirakis

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

Daniela De Silva·Nataša Pavlović·Gigliola Staffilani·Nikolaos Tzirakis

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Abstract

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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Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. 2007-02-23. Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schr\"odinger equation in $\R$. https://arxiv.org/abs/math/0702707

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