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The quintic nonlinear Schr\"odinger equation on three-dimensional Zoll manifolds
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
On the Cauchy problem for the derivative nonlinear Schroedinger equation with periodic boundary condition
It is shown that the Cauchy problem for the DNLS equation in the spatially
periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2.
Moreover, global well-posedness is shown for s \geq 1 and data with small L^2
norm.Comment: 22 page
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