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

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.