Non-uniform continuity of the solution map to the rotation-two-component Camassa-Holm system

Abstract

Studied in this paper is the rotation-two-component Camassa-Holm system, which can be considered as a model for the propagation of long-crested shallow-water waves in the equatorial ocean regions with the weak Coriolis effect due to the Earth's rotation. It is shown that the data-to-solution map is not uniformly continuous on both the line and the circle in certain Sobolev spaces depending on the effect of the Coriolis force and stretching balance parameters. The non-uniform dependence result is established by using the method of approximate solutions with refined energy estimates.