GIBBS MEASURE FOR THE FOCUSING FRACTIONAL NLS ON THE TORUS

We study the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schro\dinger equation on the multidimensional torus. We identify the sharp mass threshold for normalizability and nonnormalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz, Rose, and Speer [J. Statist. Phys., 50 (1988), pp. 657--687], Bourgain [Comm. Math. Phys., 166 (1994), pp. 1--26], and Oh, Sosoe, and Tolomeo [Invent. Math., 227 (2022), pp. 1323--1429] on the one-dimensional nonlinear Schro\dinger equations. To this purpose, we establish an almost sharp fractional Gagliardo--Nirenberg--Sobolev inequality on the torus, which is of independent interest.

Automated summary

We study the construction of Gibbs measures for the focusing mass-critical fractional nonlinear Schrödinger equation on the multidimensional torus. We identify the sharp mass threshold for normalizability and nonnormalizability of the focusing Gibbs measures, and establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus.