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arXiv · 1101.4565

The quintic nonlinear Schr\"odinger equation on three-dimensional Zoll manifolds

Sebastian Herr
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Abstract

Let (M,g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the 3-sphere S^3 with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schr\"odinger equation i\partial_t u+\Delta u=\pm|u|^4u, u|_{t=0}=u_0 is solved for small initial data u_0 in the energy space H^1(M), which is the scaling-critical space. Further, local well-posedness for large data, as well as persistence of higher initial Sobolev regularity is obtained. This extends previous results of Burq-G\'erard-Tzvetkov to the endpoint case.

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Sebastian Herr. 2011-01-24. The quintic nonlinear Schr\"odinger equation on three-dimensional Zoll manifolds. https://doi.org/10.1353/ajm.2013.0040

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