Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle

被引:0
作者
Kondo, Toshiki [1 ]
Okamoto, Mamoru [1 ]
机构
[1] Osaka Univ, Grad Sch Sci, Dept Math, Toyonaka, Osaka 5600043, Japan
来源
PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS | 2025年 / 6卷 / 02期
基金
日本学术振兴会;
关键词
Schr & ouml; dinger equation; Ill-posedness; Norm inflation; Unconditional uniqueness; INITIAL-VALUE PROBLEM; WELL-POSEDNESS; SCHRODINGER-EQUATIONS; LOCAL EXISTENCE; ILL-POSEDNESS; KDV; 3RD-ORDER; NLS;
D O I
10.1007/s42985-025-00315-4
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We consider a periodic higher-order nonlinear Schr & ouml;dinger equation with the nonlinearity uk partial derivative xu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u<^>k \partial _xu$$\end{document}, where k is a natural number. We prove the norm inflation in a subspace of the Sobolev space Hs(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>s(\mathbb {T})$$\end{document} for any s is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \mathbb {R}$$\end{document}. In particular, the Cauchy problem is ill-posed in Hs(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>s(\mathbb {T})$$\end{document} for any s is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \mathbb {R}$$\end{document}.
引用
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页数:14
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