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arXiv · 2310.06658

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

Jonathan Eckhardt·Aleksey Kostenko
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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.

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Jonathan Eckhardt, Aleksey Kostenko. 2025-05-18. The conservative Camassa-Holm flow with step-like irregular initial data. https://doi.org/10.1112/plms.70050

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