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2012 On the global well-posedness of energy-critical Schrödinger equations in curved spaces
Alexandru Ionescu, Benoit Pausader, Gigliola Staffilani
Anal. PDE 5(4): 705-746 (2012). DOI: 10.2140/apde.2012.5.705

Abstract

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrödinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig and Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (a theorem of Colliander, Keel, Staffilani, Takaoka and Tao).

As an application we prove global well-posedness and scattering in H1 for the energy-critical defocusing initial-value problem

( i t + Δ g ) u = u | u | 4 , u ( 0 ) = ϕ ,

on hyperbolic space 3.

Citation

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Alexandru Ionescu. Benoit Pausader. Gigliola Staffilani. "On the global well-posedness of energy-critical Schrödinger equations in curved spaces." Anal. PDE 5 (4) 705 - 746, 2012. https://doi.org/10.2140/apde.2012.5.705

Information

Received: 5 August 2010; Accepted: 1 April 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1264.35215
MathSciNet: MR3006640
Digital Object Identifier: 10.2140/apde.2012.5.705

Subjects:
Primary: 35Q55

Rights: Copyright © 2012 Mathematical Sciences Publishers

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