Global well-posedness for the L2 critical nonlinear Schrödinger equation in higher dimensions

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

doi:10.3934/cpaa.2007.6.1023

Global well-posedness for the L² critical nonlinear Schrödinger equation in higher dimensions

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

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The initial value problem for the L² critical semilinear Schrödinger equation in ℝ,ⁿ, n ≥ 3 is considered. We show that the problem is globally well posed in H^s(ℝ ⁿ) when 1 > s > & √7-1/3 for n = 3, and when 1 > s > -(n-2)+√(n-2)²+8(n-2)/4 for n > 4 We use the "I-method" combined with a local in time Morawetz estimate.

Original language	English (US)
Pages (from-to)	1023-1041
Number of pages	19
Journal	Communications on Pure and Applied Analysis
Volume	6
Issue number	4
DOIs	https://doi.org/10.3934/cpaa.2007.6.1023
State	Published - Dec 2007

Keywords

Global well-posedness
I-method
Morawetz estimates
Schrödinger equations

ASJC Scopus subject areas

Analysis
Applied Mathematics

Online availability

10.3934/cpaa.2007.6.1023

Library availability

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Cite this

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title = "Global well-posedness for the L2 critical nonlinear Schr{\"o}dinger equation in higher dimensions",

abstract = "The initial value problem for the L2 critical semilinear Schr{\"o}dinger equation in ℝ,n, n ≥ 3 is considered. We show that the problem is globally well posed in Hs(ℝ n) when 1 > s > & √7-1/3 for n = 3, and when 1 > s > -(n-2)+√(n-2)2+8(n-2)/4 for n > 4 We use the {"}I-method{"} combined with a local in time Morawetz estimate.",

keywords = "Global well-posedness, I-method, Morawetz estimates, Schr{\"o}dinger equations",

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year = "2007",

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AU - De Silva, Daniela

AU - Pavlović, Nataša

AU - Staffilani, Gigliola

AU - Tzirakis, Nikolaos

PY - 2007/12

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AB - The initial value problem for the L2 critical semilinear Schrödinger equation in ℝ,n, n ≥ 3 is considered. We show that the problem is globally well posed in Hs(ℝ n) when 1 > s > & √7-1/3 for n = 3, and when 1 > s > -(n-2)+√(n-2)2+8(n-2)/4 for n > 4 We use the "I-method" combined with a local in time Morawetz estimate.

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