On the global well-posedness of energy-critical Schrodinger equations in curved spaces
Author(s)Ionescu, Alexandru; Pausader, Benoit; Staffilani, Gigliola
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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 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[superscript 1] for the energy-critical defocusing initial-value problem (i∂t + Δ[subscript g])u = u|u|[superscript 4], u(0) = ϕ, on hyperbolic space ℍ[superscript 3].
Original manuscript September 8, 2010
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Analysis & PDE
Mathematical Sciences Publishers
Ionescu, Alexandru, Benoit Pausader, and Gigliola Staffilani. “On the global well-posedness of energy-critical Schrödinger equations in curved spaces.” Analysis & PDE 5, no. 4 (November 27, 2012): 705-746.