#### 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
##### Milestones
Received: 29 July 2016
Revised: 5 March 2017
Accepted: 24 April 2017
Published: 1 July 2017
##### Authors
 Fabrice Planchon Université Côte d’Azur CNRS, LJAD Parc Valrose 06108 Nice France Nikolay Tzvetkov Department of Mathematics Université de Cergy-Pontoise 2, Avenue A. Chauvin 95302 Cergy-Pontoise Cedex France Nicola Visciglia Dipartimento di Matematica Università Degli Studi di Pisa Largo Bruno Pontecorvo 5 I-56127 Pisa Italy