Analysis & PDE
- Anal. PDE
- Volume 5, Number 4 (2012), 705-746.
On the global well-posedness of energy-critical Schrödinger equations in curved spaces
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 for the energy-critical defocusing initial-value problem
on hyperbolic space .
Anal. PDE, Volume 5, Number 4 (2012), 705-746.
Received: 5 August 2010
Accepted: 1 April 2011
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Ionescu, Alexandru; Pausader, Benoit; Staffilani, Gigliola. On the global well-posedness of energy-critical Schrödinger equations in curved spaces. Anal. PDE 5 (2012), no. 4, 705--746. doi:10.2140/apde.2012.5.705. https://projecteuclid.org/euclid.apde/1513731243