A Lipschitz metric for the Camassa–Holm equation
We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is requ...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Cambridge University Press
2020
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| _version_ | 1826293102641414144 |
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| author | Carrillo, JA Grunert, K Holden, H |
| author_facet | Carrillo, JA Grunert, K Holden, H |
| author_sort | Carrillo, JA |
| collection | OXFORD |
| description | We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.
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| first_indexed | 2024-03-07T03:24:55Z |
| format | Journal article |
| id | oxford-uuid:b8b739b0-7d5d-4e8a-a86a-0d1e2bc2992e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:24:55Z |
| publishDate | 2020 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:b8b739b0-7d5d-4e8a-a86a-0d1e2bc2992e2022-03-27T04:57:46ZA Lipschitz metric for the Camassa–Holm equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b8b739b0-7d5d-4e8a-a86a-0d1e2bc2992eEnglishSymplectic ElementsCambridge University Press2020Carrillo, JAGrunert, KHolden, HWe analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric. |
| spellingShingle | Carrillo, JA Grunert, K Holden, H A Lipschitz metric for the Camassa–Holm equation |
| title | A Lipschitz metric for the Camassa–Holm equation |
| title_full | A Lipschitz metric for the Camassa–Holm equation |
| title_fullStr | A Lipschitz metric for the Camassa–Holm equation |
| title_full_unstemmed | A Lipschitz metric for the Camassa–Holm equation |
| title_short | A Lipschitz metric for the Camassa–Holm equation |
| title_sort | lipschitz metric for the camassa holm equation |
| work_keys_str_mv | AT carrilloja alipschitzmetricforthecamassaholmequation AT grunertk alipschitzmetricforthecamassaholmequation AT holdenh alipschitzmetricforthecamassaholmequation AT carrilloja lipschitzmetricforthecamassaholmequation AT grunertk lipschitzmetricforthecamassaholmequation AT holdenh lipschitzmetricforthecamassaholmequation |