Vol. 5, No. 4, 2012

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ISSN: 1948-206X (e-only)
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On the global well-posedness of energy-critical Schrödinger equations in curved spaces

Alexandru D. Ionescu, Benoit Pausader and Gigliola Staffilani

Vol. 5 (2012), No. 4, 705–746
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

(it + Δg)u = u|u|4,u(0) = ϕ,

on hyperbolic space 3.

Keywords
global well-posedness, energy-critical defocusing NLS, nonlinear Schrödinger equation, induction on energy
Mathematical Subject Classification 2000
Primary: 35Q55
Milestones
Received: 5 August 2010
Accepted: 1 April 2011
Published: 27 November 2012
Authors
Alexandru D. Ionescu
Department of Mathematics
Princeton University
Washington Road
Princeton, NJ 08544
United States
Benoit Pausader
Department of Mathematics
Brown University
151 Thayer Street
Providence, RI 02912
United States
Gigliola Staffilani
Department of Mathematics
Massachusetts Institue of Technology
77 Massachusetts Avenue, 2-246
Cambridge, MA 02139
United States