Zero-filter limit for the Camassa-Holm equation in Sobolev spaces

TL;DR

This paper proves that as the filter parameter approaches zero, solutions of the Camassa-Holm equation converge to solutions of the Burgers equation within the same Sobolev space topology, clarifying their relationship.

Contribution

It establishes the zero-filter limit of the Camassa-Holm equation as the Burgers equation in Sobolev spaces, answering an open question from prior research.

Findings

Convergence of Camassa-Holm solutions to Burgers solutions as filter parameter tends to zero

Validation of the limit in the same Sobolev topology as initial data

Clarification of the relationship between the two equations in the zero-filter limit

Abstract

The aim of this paper is to answer the question left in \cite{GL} (Math. Z. (2015) 281). We prove that the zero-filter limit of the Camassa-Holm equation is the Burgers equation in the same topology of Sobolev spaces as the initial data.