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

TL;DR

This paper investigates the continuity properties of the solution map for a two-component higher order Camassa-Holm system, establishing its well-posedness and H"older continuity in Sobolev spaces.

Contribution

It proves the well-posedness of the system in Sobolev spaces and characterizes the H"older continuity of the solution map between different Sobolev topologies.

Findings

Solution map is continuous in Sobolev spaces $H^{s} imes H^{s-2}$ for $s > 7/2$.

Solution map exhibits H"older continuity in lower Sobolev spaces with an exponent depending on $s$ and $r$.

The H"older exponent is explicitly expressed in terms of the Sobolev indices.

Abstract

This work studies the Cauchy problem of a two-component higher order Camassa-Holm system, which is well-posed in Sobolev spaces $H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$, $s>\frac{7}{2}$ and its solution map is continuous. We show that the solution map is H\"{o}lder continuous in $H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$ equipped with the $H^{r}(\mathbb{R})\times H^{r-2}(\mathbb{R})$-topology for $1\leq r<s$, and the H\"{o}lder exponent is expressed in terms of $s$ and $r$.