On the global well-posedness of energy-critical Schrödinger equations in curved spaces

Details On the global well-posedness of energy-critical Schrödinger equations in curved spaces On the global well-posedness of energy-critical Schrödinger equations in curved spaces

2010-08-06

arxiv.org/abs/1008.1237v2

Mathematics

Analysis of PDEs

43 pages, references added

Scientific paper

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger 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-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the energy-critical defocusing initial-value problem (i\partial_t+\Delta_\g)u=u|u|^{4} on the hyperbolic space $H^3$.

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