Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa-Holm equation in time weighted $H^1(\mathbb{R})$ space

TL;DR

This paper establishes the existence and uniqueness of globally conservative weak solutions for a weakly dissipative Camassa-Holm equation in a time weighted $H^1$ space, including peakon solutions.

Contribution

It introduces a new semi-linear system and auxiliary variables to prove the global existence and uniqueness of conservative solutions in the specified function space.

Findings

Existence of globally conservative weak solutions in time weighted $H^1$ space.

Peakons are shown to be conservative weak solutions.

Uniqueness of solutions is proved via auxiliary variables and semilinear system.

Abstract

In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted $H^1$ space. Second, we show that the peakon solutions are conservative weak solutions in $H^1.$ Finally, given a conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables.