On the multi-bubble blow-up solutions to focusing mass-critical stochastic nonlinear Schrödinger equations in dimensions one and two

• 成果类型: Article
• 署名作者: Su, Yiming; Zhang, Deng
• 署名单位: Hangzhou Normal University; Shanghai Jiao Tong University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01421-5
• 发表日期: 2025
• 页码: 163-246
• 关键词: schrodinger-equation; 2-bubble solutions; minimal mass; CONSTRUCTION; EXISTENCE; DYNAMICS; noise; uniqueness; STABILITY; points

摘要

We study the bubbling phenomena for focusing mass-critical nonlinear Schr & ouml;dinger equations, driven by linear multiplicative noise in the sense of controlled rough paths. In both dimensions one and two, we give a pathwise construction of stochastic multi-bubble blow-up solutions, which concentrate at finitely many distinct points, and behave asymptotically like a sum of pseudo-conformal blow-up solutions. In particular, this provides the first examples of mass quantization phenomenon in the stochastic case. It applies to the canonical deterministic model as well and complements the classical work (Merle, Comm. Math. Phys. 129(2), 223-240 (1990)). The second main result is concerned with the uniqueness of multi-bubble solutions in the energy class with the rate (T-t)3+zeta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T-t)<^>{3+\zeta }$$\end{document}, where zeta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta $$\end{document} can be any small positive constant. The uniqueness result is new in both the stochastic and deterministic cases. Via the pseudo-conformal symmetry, it also yields the corresponding uniqueness of deterministic multi-solitons. In the special single-bubble case, it provides the conditional uniqueness of stochastic critical-mass blow-up solutions, which improves the recent work (Su and Zhang, J. Funct. Anal. 284, 109796 (2023)).