On the initial value problem for the two-coupled Camassa–Holm system in Besov spaces

被引:0
作者
Haiquan Wang
Gezi Chong
机构
[1] Northwest University,School of Mathematics
来源
Monatshefte für Mathematik | 2020年 / 193卷
关键词
The two-coupled Camassa–Holm system; Non-uniformly continuous dependence; Hölder continuity; Besov spaces; 35B30; 35G25;
D O I
暂无
中图分类号
学科分类号
摘要
Considered herein is the Cauchy problem for the two-coupled Camassa–Holm system. Based on the local well-posedness results for this problem, it is shown that the solution map z0↦z(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{0}\mapsto z(t)$$\end{document} of this problem in the periodic case is not uniformly continuous in Besov spaces Bp,rs(T)×Bp,rs(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{s}_{p,r}(\mathbb {T})\times B^{s}_{p,r}(\mathbb {T}) $$\end{document} with s>max{3/2,1+1/p},1≤p,r≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>\max \{3/2,1+1/p\}, 1\le p,r\le \infty $$\end{document} by using the method of approximate solutions. In the non-periodic case, the non-uniform continuity of this solution map in Besov spaces B2,rs(R)×B2,rs(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{s}_{2,r}(\mathbb {R})\times B^{s}_{2,r}(\mathbb {R}) $$\end{document} with s>3/2,2≤r≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>3/2, 2\le r\le \infty $$\end{document} is established. Finally, the Hölder continuity of the solution map in Besov spaces is proved.
引用
收藏
页码:479 / 505
页数:26
相关论文
共 39 条
  • [21] Non-uniform continuity on initial data for a Camassa-Holm-type equation in Besov space
    Wu, Xing
    Xiao, Yu
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2021, 494 (02)
  • [22] Non-uniform Dependence on Initial Data for the Generalized Camassa-Holm-Novikov Equation in Besov Space
    Wu, Xing
    Yu, Yanghai
    Xiao, Yu
    JOURNAL OF MATHEMATICAL FLUID MECHANICS, 2021, 23 (04)
  • [23] WELL-POSEDNESS FOR A MODIFIED TWO-COMPONENT CAMASSA-HOLM SYSTEM IN CRITICAL SPACES
    Yan, Kai
    Yin, Zhaoyang
    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2013, 33 (04) : 1699 - 1712
  • [24] The periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system
    Zhang, Lei
    Qiao, Zhijun
    JOURNAL OF DIFFERENTIAL EQUATIONS, 2020, 268 (03) : 1270 - 1305
  • [25] A new generalised two-component Camassa–Holm type system with waltzing peakons and wave breaking
    Li Yang
    Baoshuai Zhang
    Shouming Zhou
    Nonlinear Differential Equations and Applications NoDEA, 2018, 25
  • [26] A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons
    Xiaolin Pan
    Shouming Zhou
    Zhijun Qiao
    Journal of Nonlinear Mathematical Physics, 2023, 30 : 1153 - 1189
  • [27] On the Cauchy problem for the fractional drift- diffusion system in critical Besov spaces
    Zhao, Jihong
    Liu, Qiao
    APPLICABLE ANALYSIS, 2014, 93 (07) : 1431 - 1450
  • [28] A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons
    Pan, Xiaolin
    Zhou, Shouming
    Qiao, Zhijun
    JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, 2023, 30 (03) : 1153 - 1189
  • [29] A new generalised two-component Camassa-Holm type system with waltzing peakons and wave breaking
    Yang, Li
    Zhang, Baoshuai
    Zhou, Shouming
    NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 2018, 25 (04):
  • [30] Blow-up data for a two-component Camassa-Holm system with high order nonlinearity
    Wang, Zhaopeng
    Yan, Kai
    JOURNAL OF DIFFERENTIAL EQUATIONS, 2023, 358 : 256 - 294