On the growth of high Sobolev norms of the cubic nonlinear Schr\"odinger equation on $\mathbb{R}\times \mathbb{T}$

TL;DR

This paper establishes polynomial bounds on the growth of high Sobolev norms for solutions to the cubic nonlinear Schrödinger equation on the product manifold of real line and torus, using an iterative approach inspired by Bourgain.

Contribution

It provides the first polynomial growth bounds for high Sobolev norms of this equation on $\

Findings

Polynomial bounds on Sobolev norm growth are achieved.

The method adapts Bourgain's iteration technique to the product manifold setting.

Results contribute to understanding long-time behavior of solutions.

Abstract

We consider the cubic nonlinear Schr\"odinger equation on product manifolds $\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of the solutions. The main ingredient of the proof is to establish an iteration bound, which is based on the idea used by Bourgain in \cite{B1}.