Existence and Uniqueness of Global Weak solutions of the Camassa-Holm Equation with a Forcing

TL;DR

This paper establishes the existence and uniqueness of global weak solutions for the forced Camassa-Holm equation in the Sobolev space $H^1( eal)$, addressing challenges posed by the forcing term.

Contribution

It introduces new estimates and a balance law approach to prove well-posedness of weak solutions with forcing, extending prior results without forcing.

Findings

Proved existence of global weak solutions in $H^1( eal)$

Established uniqueness of these solutions

Addressed the impact of forcing on properties like conservation laws

Abstract

In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.