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

• 成果类型: Article
• 署名作者: Oh, Tadahiro; Sosoe, Philippe; Tolomeo, Leonardo
• 署名单位: University of Edinburgh; University of Edinburgh; Heriot Watt University; Cornell University; University of Bonn
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01080-y
• 发表日期: 2022
• 页码: 1323-1429
• 关键词: nonlinear schrodinger-equation; blow-up solutions; global well-posedness; data cauchy-theory; invariant-measures; wave-equation; ground-state; statistical-mechanics; DYNAMICS; mass

摘要

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrodinger equation on the one-dimensional torus. In an influential paper, Lebowitz et al. (J Stat Phys 50(3-4):657-687, 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 (Ann Inst H Poincare Anal Non Lineaire 31(6):1267-1288, 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 et al. (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrodinger equation on the one-dimensional torus.