Non-uniform dependence of the data-to-solution map for the Hunter–Saxton equation in Besov spaces

The Cauchy problem for the Hunter–Saxton equation is known to be locally well posed in Besov spaces B2,rs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^s_{2,r} $$\end{document} on the circle. We prove that the data-to-solution map is not uniformly continuous from any bounded subset of B2,rs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^s_{2,r} $$\end{document} to C([0,T];B2,rs)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C([0, T]; B^s_{2,r} )$$\end{document}. We also show that the solution map is Hölder continuous with respect to a weaker topology.