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Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear SchrÖdinger Equations

N. Burq·P. Gerard·N. Tzvetkov
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Abstract

We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold $M$. We prove global existence of strong $H^1$ solutions on $M=S^3$ and $M=S^2\times S^1$ 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 $\R^3$ and the flat torus $\T^3$ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.

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N. Burq, P. Gerard, N. Tzvetkov. 2005-01-27. Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear SchrÖdinger Equations. https://arxiv.org/abs/math/0409015

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