Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear SchrÖdinger Equations
N. Burq, P. Gerard, N. Tzvetkov
Abstract
We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold M. We prove global existence of strong H1 solutions on M=S3 and M=S2× S1 as far as the nonlinearity is defocusing and sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space 3 and the flat torus 3 respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.
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