#### Vol. 10, No. 5, 2017

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On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds

### Fabrice Planchon, Nikolay Tzvetkov and Nicola Visciglia

Vol. 10 (2017), No. 5, 1123–1147
##### Abstract

Using suitable modified energies, we study higher-order Sobolev norms’ growth in time for the nonlinear Schrödinger equation (NLS) on a generic 2- or 3-dimensional compact manifold. In two dimensions, we extend earlier results that dealt only with cubic nonlinearities, and get polynomial-in-time bounds for any higher-order nonlinearities. In three dimensions, we prove that solutions to the cubic NLS grow at most exponentially, while for the subcubic NLS we get polynomial bounds on the growth of the ${H}^{2}$ norm.

##### Keywords
growth of Sobolev norms, NLS on compact manifolds
Primary: 35Q55