Uniqueness of dissipative solutions for the Camassa-Holm equation
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912002381709312 |
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| author | Grunert, Katrin |
| author_facet | Grunert, Katrin |
| contents | We show that the Cauchy problem for the Camassa-Holm equation has a unique, global, weak, and dissipative solution for any initial data $u_0\in H^1(\mathbb{R})$, such that $u_{0,x}$ is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15344 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniqueness of dissipative solutions for the Camassa-Holm equation Grunert, Katrin Analysis of PDEs 35A02, 35L45, 35B60 We show that the Cauchy problem for the Camassa-Holm equation has a unique, global, weak, and dissipative solution for any initial data $u_0\in H^1(\mathbb{R})$, such that $u_{0,x}$ is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution. |
| title | Uniqueness of dissipative solutions for the Camassa-Holm equation |
| topic | Analysis of PDEs 35A02, 35L45, 35B60 |
| url | https://arxiv.org/abs/2311.15344 |