Global conservative solution for the periodic $\mu$-Camassa-Holm equation

TL;DR

This paper establishes the existence of global conservative solutions for the periodic -mu-Camassa-Holm equation using a coordinate transformation to Lagrangian coordinates, ensuring continuous dependence on initial data and a semigroup property.

Contribution

It introduces a novel approach with coordinate transformation to Lagrangian coordinates to prove global conservative solutions for the periodic -mu-Camassa-Holm equation.

Findings

Existence of global conservative solutions proven.

Solutions depend continuously on initial data.

Solutions exhibit a semigroup property.

Abstract

In this paper we mainly investigate the periodic $\mu$-Camassa-Holm equation. We show the existence of global conservative solutions to the Cauchy problem of the periodic $\mu$-Camassa-Holm equation. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. Our solutions depend continuously on the initial data and has a semigroup property.