The energy-critical nonlinear Schrödinger equation on a product of spheres

Let $(M,g)$ be a compact smooth $3$-dimensional Riemannian manifold without boundary. It is proved that the energy-critical nonlinear Schrödinger equation is globally well-posed for small initial data in $H^1(M)$, provided that a certain tri-linear estimate for free solutions holds true. This estimate is known to hold true on the sphere and tori in $3d$ and verified here in the case $\mathbb{S} \times \mathbb{S}^2$. The necessity of a weak form of this tri-linear estimate is also discussed.

Keywords

nonlinear Schrödinger equation, compact manifold, wellposedness

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35R01.

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Accepted 16 September 2014

Published 20 May 2015