Global solutions for 1D cubic dispersive equations, part III: the quasilinear Schrödinger flow

• 成果类型: Article
• 署名作者: Ifrim, Mihaela; Tataru, Daniel
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01356-7
• 发表日期: 2025
• 页码: 221-304
• 关键词: nonlinear schrodinger-equation; local well-posedness; 2nd-order hyperbolic operators; initial-value problem; nonsmooth coefficients; normal forms; rough solutions; cauchy-problem; large time; life-span

摘要

The first target of this article is the local well-posedness question for 1D quasilinear Schr & ouml;dinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a sharp local well-posedness result. The second goal of this article is to consider the long-time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions; the conjecture was initially proved for a class of semilinear Schr & ouml;dinger type models. Our work here establishes the above conjecture for 1D quasilinear Schr & ouml;dinger flows. Precisely, we show that if the problem has phase rotation symmetry and is conservative and defocusing, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows where no localization condition is imposed on the data. Furthermore, we prove the global well-posedness at the minimal Sobolev regularity as in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that small,& varepsilon;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon $\end{document}size data yields long-time solutions on the & varepsilon;-8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon <^>{-8}$\end{document}time-scale. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schr & ouml;dinger flows, also at minimal regularity.