Non-uniform dependence on initial data for the periodic modified Camassa–Holm equation

In this paper, we study the periodic Cauchy problem for the modified Camassa–Holm equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_t+um_x+2u_xm=0,\quad m=(1-\partial_x^2)^2u$$\end{document}, and show that the solution map is not uniformly continuous in Sobolev spaces \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 s > 7/2. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.