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

Abstract

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger 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.

29 pages, improved result

References in corpus (2)

• Global well-posedness for Schrödinger equations with derivative
• Minimal-mass blowup solutions of the mass-critical NLS

Cited by in corpus (5)

• Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^2$
• The low regularity global solutions for the critical generalized KdV equation
• Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{ö}dinger equation in higher dimensions
• Global well-posedness for the defocusing, quintic nonlinear Schrödinger equation in one dimension
• On the mass-critical generalized KdV equation