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arXiv · 1403.7965

The energy-critical nonlinear Schr\"odinger equation on a product of spheres

Sebastian Herr·Nils Strunk
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Abstract

Let $(M,g)$ be a compact smooth $3$-dimensional Riemannian manifold without boundary. It is proved that the energy-critical nonlinear Schr\"odinger 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.

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Sebastian Herr, Nils Strunk. 2014-03-31. The energy-critical nonlinear Schr\"odinger equation on a product of spheres. https://doi.org/10.4310/mrl.2015.v22.n3.a7

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