The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

DOI

10.2991/jnmp.2003.10.s1.2How to use a DOI?

Abstract

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 arbirary 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.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Nicolas Burq
AU  - Patrick Gérard
AU  - Nikolay Tzvetkov
PY  - 2003
DA  - 2003/08/01
TI  - The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold
JO  - Journal of Nonlinear Mathematical Physics
SP  - 12
EP  - 27
VL  - 10
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2003.10.s1.2
DO  - 10.2991/jnmp.2003.10.s1.2
ID  - Burq2003
ER  -