The conservative Camassa-Holm flow with step-like irregular initial data
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910948434903040 |
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| author | Eckhardt, Jonathan Kostenko, Aleksey |
| author_facet | Eckhardt, Jonathan Kostenko, Aleksey |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_06658 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The conservative Camassa-Holm flow with step-like irregular initial data Eckhardt, Jonathan Kostenko, Aleksey Analysis of PDEs Spectral Theory Primary 37K15, 34L05, Secondary 34A55, 35Q51 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. |
| title | The conservative Camassa-Holm flow with step-like irregular initial data |
| topic | Analysis of PDEs Spectral Theory Primary 37K15, 34L05, Secondary 34A55, 35Q51 |
| url | https://arxiv.org/abs/2310.06658 |