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

Eckhardt, Jonathan (author), Kostenko, Aleksej Sergejevič (author)

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.

• Language: English
• Keywords: conservative Camassa-Holm flow, inverse spectral transform
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.05.2025
• Year: 2025
• Number of pages: 42 str.
• Numbering: Vol. 130, iss. 5, article no. e70050
• PID: 20.500.12556/RUL-175177
• UDC: 517.9
• ISSN on article: 0024-6115
• DOI: 10.1112/plms.70050
• COBISS.SI-ID: 236509955
• Publication date in RUL: 20.10.2025

Record is a part of a journal

• Title: Proceedings of the London Mathematical Society
• Shortened title: Proc. Lond. Math. Soc.
• Publisher: Wiley, London Mathematical Society
• ISSN: 0024-6115
• COBISS.SI-ID: 26179584

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: FWF - Austrian Science Fund
• Project number: I4600

• Funder: ARRS - Slovenian Research Agency
• Project number: N1-0137
• Name: Nelinearni Valovi in Spektralna Teorija