Lipschitz metric for the modified two-component Camassa-Holm system

TL;DR

This paper establishes the existence and Lipschitz continuity of global weak solutions for the modified two-component Camassa-Holm system using a coordinate transformation and novel distance functions.

Contribution

It introduces a new approach with coordinate transformation and distance functions to prove global weak solutions for the modified two-component Camassa-Holm system.

Findings

Proved existence of global conservative weak solutions.

Established Lipschitz continuity of solutions.

Utilized energy density and Radon measures in analysis.

Abstract

This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa-Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.