Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr\"Odinger Equations

N. Burq; P. Gerard; N. Tzvetkov

Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr\"Odinger Equations

Analysis of PDEs 2007-05-23 v2

Authors: N. Burq , P. Gerard , N. Tzvetkov

Abstract

We study nonlinear Schr\"odinger 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.

Keywords

nonlinear schrödinger equation schrödinger equation nonlinear schrödinger equations

Cite

@article{arxiv.math/0409015,
  title  = {Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr\"Odinger Equations},
  author = {N. Burq and P. Gerard and N. Tzvetkov},
  journal= {arXiv preprint arXiv:math/0409015},
  year   = {2007}
}

Comments

Lemma 4.6 in the previous version was false, we made slight modifications to use only a weaker version of this lemma

Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr\"Odinger Equations

Abstract

Keywords

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