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 ${\Bbb S}^3$ 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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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1271-1290
Launched on MUSE
2013-09-25
Open Access
No
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