Gibbs measure for the focusing fractional NLS on the torus
arXiv:2109.05626 · doi:10.1137/21M1445946
Abstract
We study the construction of the Gibbs measures for the {\it focusing} mass-critical fractional nonlinear Schrödinger 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 Schrödinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.
23 pages