A Lipschitz metric for the Hunter–Saxton equation
We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniquenes...
| Main Authors: | , , |
|---|---|
| Format: | Journal article |
| Published: |
Taylor & Francis
2019
|
| _version_ | 1826295650262712320 |
|---|---|
| author | Carrillo de la Plata, JA Grunert, K Holden, H |
| author_facet | Carrillo de la Plata, JA Grunert, K Holden, H |
| author_sort | Carrillo de la Plata, JA |
| collection | OXFORD |
| description | We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS 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 article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric. |
| first_indexed | 2024-03-07T04:04:19Z |
| format | Journal article |
| id | oxford-uuid:c59fe1df-fab1-4054-8f43-a3cc67fc3c58 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:04:19Z |
| publishDate | 2019 |
| publisher | Taylor & Francis |
| record_format | dspace |
| spelling | oxford-uuid:c59fe1df-fab1-4054-8f43-a3cc67fc3c582022-03-27T06:32:21ZA Lipschitz metric for the Hunter–Saxton equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c59fe1df-fab1-4054-8f43-a3cc67fc3c58Symplectic ElementsTaylor & Francis2019Carrillo de la Plata, JAGrunert, KHolden, HWe analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS 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 article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric. |
| spellingShingle | Carrillo de la Plata, JA Grunert, K Holden, H A Lipschitz metric for the Hunter–Saxton equation |
| title | A Lipschitz metric for the Hunter–Saxton equation |
| title_full | A Lipschitz metric for the Hunter–Saxton equation |
| title_fullStr | A Lipschitz metric for the Hunter–Saxton equation |
| title_full_unstemmed | A Lipschitz metric for the Hunter–Saxton equation |
| title_short | A Lipschitz metric for the Hunter–Saxton equation |
| title_sort | lipschitz metric for the hunter saxton equation |
| work_keys_str_mv | AT carrillodelaplataja alipschitzmetricforthehuntersaxtonequation AT grunertk alipschitzmetricforthehuntersaxtonequation AT holdenh alipschitzmetricforthehuntersaxtonequation AT carrillodelaplataja lipschitzmetricforthehuntersaxtonequation AT grunertk lipschitzmetricforthehuntersaxtonequation AT holdenh lipschitzmetricforthehuntersaxtonequation |