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

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.