Sharp well-posedness and ill-posedness of a higher-order modified Camassa-Holm equation. (Q1941792)

The well-posedness and ill-posedness of the following modified Camassa-Holm equation \[ \partial _t u + \partial _x^{2n+1}u + \frac {1}{2} \partial _x( u^2)+(\text{id} - \partial _x^2)^{-1} \partial _x \left [ u^2 + \frac {1}{2}( \partial _x u)^2 \right ] = 0, \quad (t,x)\in (0,\infty )\times \mathbb {R}, \] with initial condition \(u_0\) are studied for integers \(n\geq 2\). It is shown that, given \(u_0\in H^{-n+(5/4)}(\mathbb {R})\) small enough, the above Cauchy problem is well-posed, thereby extending the well-posedness in \(H^s(\mathbb {R})\) established in [\textit{Y. S. Li, W. Yan} and \textit{X. Y. Yang}, J. Evol. Equ. 10, No. 2, 465--486 (2010; Zbl 1239.35141)] for \(s>-n+(5/4)\). It is also proved that, if \(s<-n+(5/4)\), the map \(u_0\longmapsto [t\mapsto u(t)]\) associating the solution to the modified Camassa-Holm equation to its initial condition cannot be \(C^2\)-smooth at zero as a map from \(\dot {H}^s(\mathbb {R})\) to \(C([0,T];\dot {H}^s(\mathbb {R}))\).