On the growth of high Sobolev norms of the cubic nonlinear Schrödinger equation on $\mathbb R\times\mathbb T$

Abstract

We consider the cubic defocusing nonlinear Schrödinger equation on product manifold $\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of the solutions, which measure the transfer of energy from low to high modes as time grows on. The main ingredient of the proof is to establish an iteration bound, which is based on the idea of Bourgain in [3].