Two-component system modelling shallow-water waves with constant vorticity under the Camassa-Holm scaling

TL;DR

This paper derives a new two-component shallow-water wave model with constant vorticity from Green-Naghdi equations, analyzes its well-posedness, blow-up behavior, and conditions for global solutions under the Camassa-Holm scaling.

Contribution

It introduces a novel two-component system incorporating vorticity effects, differing from previous models, and provides mathematical analysis of its well-posedness and solution behavior.

Findings

Established local well-posedness in Besov spaces.

Presented a blow-up criterion for solutions.

Provided a sufficient condition for global strong solutions.

Abstract

This paper is concerned with the derivation of a two-component system modelling shallow-water waves with constant vorticity under the Camassa-Holm scaling from our newly established Green-Naghdi equations with a linear shear. It is worth pointing out that the $\rho$ component in this new system is quite different from the previous two-component system due to the effects of both vorticity and larger amplitude. We then establish the local well-posedness of this new system in Besov spaces, and present a blow-up criterion. We finally give a sufficient condition for global strong solutions to the system in some special case.