Periodic nonlinear Schr\"odinger equation with distributional potential and invariant measures

TL;DR

This paper extends the analysis of the periodic nonlinear Schrödinger equation by incorporating a distributional potential, proving global well-posedness and invariance of measures for low regularity data.

Contribution

It introduces new Strichartz estimates for NLS with distributional potentials, establishing global well-posedness and measure invariance in this setting.

Findings

Global well-posedness for initial data of full measure

Invariance of the Gibbs measure under the flow

Extension of previous results to distributional potentials

Abstract

In this paper, we continue some investigations on the periodic NLSE started by Lebowitz, Rose and Speer and by Bourgain with the addition of a distributional multiplicative potential. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measure, after suitable truncation in the focusing case. The set and the measure are invariant under the flow. The main ingredients used are Strichartz estimates on periodic NLS with distributional potential to obtain local well-posedness for low regularity initial data.