Uniqueness of Dissipative Solution for Camassa-Holm Equation with Peakon-Antipeakon Initial Data

TL;DR

This paper proves the uniqueness of dissipative solutions for the Camassa-Holm equation with specific peakon-antipeakon initial data, confirming that different existing frameworks yield the same solution in finite time blowup scenarios.

Contribution

It extends the uniqueness results for dissipative solutions to the Camassa-Holm equation, aligning two major solution frameworks for a class of initial data.

Findings

Uniqueness of dissipative solutions established for specific initial data.

Different solution frameworks produce identical solutions.

Results apply to finite time gradient blowup scenarios.

Abstract

We give a proof for the uniqueness of dissipative solution for the Camassa-Holm equation with some peakon-antipeakon initial data following Dafermos' earlier resut in [5] on the Hunter-Saxton equation. Our result shows that two existing global existence frameworks, through the vanishing viscosity method by Xin-Zhang in [11] and the transformation of coordinate method for dissipative solutions by Bressan-Constantin in [3], give the same solution, for a special but typical initial data forming finite time gradient blowup.