Gibbs measure for the focusing fractional NLS on the torus

TL;DR

This paper constructs and analyzes focusing Gibbs measures for the fractional nonlinear Schrödinger equation on a torus, identifying the critical mass threshold for measure normalizability and establishing a key fractional inequality.

Contribution

It generalizes previous one-dimensional results to multi-dimensional tori and introduces an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality.

Findings

Identified the sharp mass threshold for Gibbs measure normalizability.

Established an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality.

Extended the theory of Gibbs measures to multi-dimensional fractional NLS.

Abstract

We study the construction of the Gibbs measures for the {\it focusing} mass-critical fractional nonlinear Schr\"odinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schr\"odinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.