Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus
Abstract
We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing masscritical nonlinear Schrödinger equation on the onedimensional 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 twodimensional 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 onedimensional 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 Schrödinger equation on the onedimensional torus.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1709.02045
 Bibcode:
 2017arXiv170902045O
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 60H40;
 60H30;
 35Q55
 EPrint:
 81 pages. Minor typos corrected. Published in Invent. Math