On blow up for the energy super critical defocusing nonlinear Schrodinger equations

• 成果类型: Article
• 署名作者: Merle, Frank; Raphael, Pierre; Rodnianski, Igor; Szeftel, Jeremie
• 署名单位: CY Cergy Paris Universite; Princeton University; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Sorbonne Universite; Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01067-9
• 发表日期: 2022
• 页码: 247-413
• 关键词: global well-posedness; self-similar blowup; finite-time blowup; semiclassical limit; wave equations; cauchy-problem; ii blowup; SCATTERING; DYNAMICS; REGULARITY

摘要

We consider the energy supercritical defocusing nonlinear Schrodinger equation i partial derivative(t)u + Delta u - u vertical bar u vertical bar(p-1) = 0 in dimension d >= 5. In a suitable range of energy supercritical parameters (d, p), we prove the existence of C-infinity well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blowup mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of C-infinity spherically symmetric self similar solutions to the compressible Euler equationwhose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.