Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.06489 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918281289400320 |
|---|---|
| author | Liu, Xin Thomas, Marita Titi, Edriss S. |
| author_facet | Liu, Xin Thomas, Marita Titi, Edriss S. |
| contents | We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation functional. Our study highlights that the interplay of the advection term with the stresses given by the multivalued 1-Laplacian is a crucial feature of this model. Eventhough it is an interesting model in itsef, it can also be regarded as a one-dimensional version of the momentum balance of Hibler's model for sea-ice dynamics. Therein, the stress tensor is given by a term with similar properties as the 1-Laplacian in order to account for plastic effects of the ice. For our analysis we start out from a viscoplastic Burgers equation, i.e., a suitably regularized version of the plastic Burgers equation with a small regularization parameter $\varepsilon>0$. For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit $\varepsilon\to0$ we deduce the existence of a BV-solution for the plastic Burgers equation. In addition we show that the term arising as the limit of the regularized stresses is indeed related to an element of the subdifferential of the total variation functional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06489 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics Liu, Xin Thomas, Marita Titi, Edriss S. Analysis of PDEs 35D30, 35D40, 35Q35, 35Q86, 49J45, 49J53, 76A05, 86A05 We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation functional. Our study highlights that the interplay of the advection term with the stresses given by the multivalued 1-Laplacian is a crucial feature of this model. Eventhough it is an interesting model in itsef, it can also be regarded as a one-dimensional version of the momentum balance of Hibler's model for sea-ice dynamics. Therein, the stress tensor is given by a term with similar properties as the 1-Laplacian in order to account for plastic effects of the ice. For our analysis we start out from a viscoplastic Burgers equation, i.e., a suitably regularized version of the plastic Burgers equation with a small regularization parameter $\varepsilon>0$. For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit $\varepsilon\to0$ we deduce the existence of a BV-solution for the plastic Burgers equation. In addition we show that the term arising as the limit of the regularized stresses is indeed related to an element of the subdifferential of the total variation functional. |
| title | Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics |
| topic | Analysis of PDEs 35D30, 35D40, 35Q35, 35Q86, 49J45, 49J53, 76A05, 86A05 |
| url | https://arxiv.org/abs/2601.06489 |