Uniqueness of dissipative solutions for the Camassa–Holm equation

Abstract

We show that the Cauchy problem for the Camassa–Holm equation has a unique, global, weak, and dissipative solution for any initial data $u_0\in H^1\left(R\right)$, such that $u_{0,x}$ is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution.