Uniqueness of dissipative solution for Camassa–Holm equation with peakon–antipeakon initial data

Abstract

We give a proof for the uniqueness of dissipative solution for Camassa–Holm equation with some peakon–antipeakon initial data. Our result shows that two existing global existence frameworks, through the vanishing viscosity method by Xin–Zhang in Xin and Zhang (2000) and the transformation of coordinate method for dissipative solutions by Bressan–Constantin in Bressan and Constantin (2007), give the same solution, for a special but typical initial data forming finite time gradient blowup.