Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

TL;DR

This paper establishes the optimal mass threshold for the normalizability of Gibbs measures linked to the focusing mass-critical NLS on the torus, confirming longstanding conjectures and extending results to the 2D radial case.

Contribution

It proves the optimality of the critical mass threshold for Gibbs measure normalizability in 1D and 2D, resolving open questions and confirming previous hypotheses.

Findings

Gibbs measure is normalizable at the critical mass threshold in 1D.

The critical mass threshold matches the ground state mass on the real line.

The proof applies to the 2D radial problem on the unit disc.

Abstract

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schr\"odinger equation on the one-dimensional torus. In an influential paper, Lebowitz, Rose, and Speer (1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schr\"odinger equation on the one-dimensional torus.