On the global existence and wave-breaking criteria for the two-component Camassa–Holm system

Considered herein is a two-component Camassa–Holm system modeling shallow water waves moving over a linear shear flow. A wave-breaking criterion for strong solutions is determined in the lowest Sobolev space Hs, by using the localization analysis in the transport equation theory. Moreover, an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this Sobolev space Hs.