The conservative Camassa-Holm flow with step-like irregular initial data

TL;DR

This paper extends the inverse spectral transform for the conservative Camassa-Holm flow to initial data with mixed decay and boundedness conditions, introducing new conservation laws and broadening understanding of the flow's well-posedness.

Contribution

It develops an extended inverse spectral transform for the Camassa-Holm flow with step-like initial data and uncovers new conservation laws linked to Besov norms.

Findings

Extended inverse spectral transform for step-like data

Discovery of new conservation laws for Camassa-Holm equation

Analysis of well-posedness under mild boundedness conditions

Abstract

We extend the inverse spectral transform for the conservative Camassa-Holm flow on the line to a class of initial data that requires strong decay at one endpoint but only mild boundedness-type conditions at the other endpoint. The latter condition appears to be close to optimal in a certain sense for the well-posedness of the conservative Camassa-Holm flow. As a byproduct of our approach, we also find a family of new (almost) conservation laws for the Camassa-Holm equation, which could not be deduced from its bi-Hamiltonian structure before and which are connected to certain Besov-type norms (however, in a rather involved way). These results appear to be new even under positivity assumptions on the corresponding momentum, in which case the conservative Camassa-Holm flow coincides with the classical Camassa-Holm flow and no blow-ups occur.