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 H2 norm.

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Keywords
growth of Sobolev norms, NLS on compact manifolds
Mathematical Subject Classification 2010
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