New bounds on the high Sobolev norms of the 1d NLS solutions

TL;DR

This paper develops new modified energy techniques to establish upper bounds on high Sobolev norms of solutions to the 1D periodic nonlinear Schrödinger equation, simplifying and extending previous results.

Contribution

It introduces a flexible method using modified energies, providing a simpler proof and extending bounds to higher order nonlinearities in 1D NLS.

Findings

Established upper bounds for high Sobolev norms of 1D NLS solutions.

Extended Bourgain's bounds to higher order nonlinearities.

Provided a simpler proof using integration by parts and dispersive estimates.

Abstract

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the $1$D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.