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

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 H 1 for the energy-critical defocusing initial-value problem .i@ t C g /u D ujuj 4 ; u.0/ D ;hyperbolic space ‫ވ‬ 3 .1. Introduction 705 2. Preliminaries 709 3. Proof of the main theorem 718 4. Euclidean approximations 721 5. Profile decomposition in hyperbolic spaces 725 6. Proof of Proposition 3.4 735 References 744

Locations

• Analysis & PDE - View - PDF
• arXiv (Cornell University) - View - PDF
• DSpace@MIT (Massachusetts Institute of Technology) - View - PDF
• Project Euclid (Cornell University) - View - PDF