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

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.