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### Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb{T}^3)$

Sebastian Herr, Daniel Tataru, and Nikolay Tzvetkov
Source: Duke Math. J. Volume 159, Number 2 (2011), 329-349.

#### Abstract

A refined trilinear Strichartz estimate for solutions to the Schrödinger equation on the flat rational torus $\mathbb{T}^3$ is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic nonlinear Schrödinger equation in $H^s(\mathbb{T}^3)$ for all $s\geq 1$. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

First Page:
Primary Subjects: 35Q55
Secondary Subjects: 35B33
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1312481492
Digital Object Identifier: doi:10.1215/00127094-1415889
Mathematical Reviews number (MathSciNet): MR2824485

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