Global wellposedness of cubic Camassa–Holm equations

In this paper, we study the Cauchy problem of cubic Camassa–Holm equation. This equation is derived from a model of shallow water dynamics. We prove the global existence of entropy weak solution to this problem in space H1H1 with its derivative in BVBV. The stability and uniqueness of entropy weak solution are obtained in W1,1W1,1.