Mathematics

We present a joint work with F. Planchon and N. Tzvetkov. Our aim is to get some upper bounds on the growth of higher order Sobolev norms for solutions to the nonlinear Schroedinger equation posed on compact manifolds in dimensions 2 and 3. The basic tool is the introduction of suitable modified energies that allow us on one hand to get an exponential bound in 3-d (hence improving the double exponential bound previousely proved by Burq-Gerard-Tzvetkov in 2002); on the other hand in 2-d we extend (and at the same time we provide an elementary proof of) a pioneering paper by Bourgain in 1996. If there is time we also present recent extensions to the dynamics of NLS perturbed by the harmonic oscillator.