On the growth of Sobolev norms for NLS on $2d$ and $3d$ manifolds

TL;DR

This paper investigates the long-term growth of Sobolev norms for nonlinear Schrödinger equations on 2D and 3D compact manifolds, extending previous results and providing bounds for various nonlinearities.

Contribution

It extends earlier 2D results to higher nonlinearities and establishes growth bounds for 3D NLS, including exponential and polynomial estimates.

Findings

Polynomial bounds for higher order nonlinearities in 2D

Exponential growth bound for cubic NLS in 3D

Polynomial growth bounds for sub-cubic NLS in 3D

Abstract

Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schr\"odinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic nonlinearities, and get polynomial in time bounds for any higher order nonlinearities. In $3d$, we prove that solutions to the cubic NLS grow at most exponentially, while for sub-cubic NLS we get polynomial bounds on the growth of the $H^2$-norm.