The Defocusing Quintic NLS in Four Space Dimensions
Abstract
We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
Recommended Citation
B. Dodson et al., "The Defocusing Quintic NLS in Four Space Dimensions," Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, vol. 34, no. 3, pp. 759 - 787, Elsevier, May 2017.
The definitive version is available at https://doi.org/10.1016/j.anihpc.2016.05.004
Department(s)
Mathematics and Statistics
Keywords and Phrases
Nonlinear equations; Scattering; Defocusing; Dinger equation; Four dimensions; Interaction Morawetz inequality; Localized interaction; Logarithmic failure; Quintic; Space dimensions; Sobolev spaces; Concentration compactness; Interaction Morawetz inequality; Nonlinear Schrödinger equation
International Standard Serial Number (ISSN)
0294-1449
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2017 Elsevier, All rights reserved.
Publication Date
01 May 2017
Comments
B. D. was supported by NSF grant DMS-1500424. C. M. was supported by the NSFC under grant No. 11171033 and 11231006. J. M. was supported by the NSF Postdoctoral Fellowship DMS-1400706. J. Z. was partly supported by the European Research Council, ERC-2012-ADG, project number 320845: Semi-Classical Analysis of Partial Differential Equations. We thank the anonymous referee for their helpful suggestions.