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arXiv · 1008.1237

On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces

Alexandru D. Ionescu·Benoit Pausader·Gigliola Staffilani
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Abstract

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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Alexandru D. Ionescu, Benoit Pausader, Gigliola Staffilani. 2010-08-06. On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces. https://arxiv.org/abs/1008.1237

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