Continuity properties of the data-to-solution map for the two-component higher order Camassa-Holm system

This work studies the Cauchy problem of a two-component higher order Camassa-Holm system, which is well-posed in Sobolev spaces Hs(R)×Hs−2(R), s>72 and its solution map is continuous. We show that the solution map is Hölder continuous in Hs(R)×Hs−2(R) equipped with the Hr(R)×Hr−2(R)-topology for 1≤r