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The Cauchy problem for the nonlinear Schrödinger equation on a compact manifold. (English) Zbl 1362.35282

Summary: We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbitrary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other hand, on specific manifolds such as spheres, new instability phenomena are displayed, leading to some kind of illposednesss in higher dimensions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35L15 Initial value problems for second-order hyperbolic equations
35R01 PDEs on manifolds
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