Global solutions for the two-component Camassa-Holm system

TL;DR

This paper establishes the existence of global conservative solutions for the two-component Camassa-Holm system with nonvanishing asymptotics, and links these solutions to the scalar Camassa-Holm equation through a vanishing density limit.

Contribution

It provides the first proof of global conservative solutions for the 2CH system with nonvanishing asymptotics and introduces a novel approach to derive solutions for the scalar Camassa-Holm equation.

Findings

Existence of global conservative solutions for 2CH with nonvanishing asymptotics.

Solutions are smooth if the density stays away from zero.

Solutions of 2CH have infinite propagation speed, but singularities move finitely.

Abstract

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa-Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.