Non-preservation of $α$-concavity for the porous medium equation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914656607535104 |
|---|---|
| author | Chau, Albert Weinkove, Ben |
| author_facet | Chau, Albert Weinkove, Ben |
| contents | We show that the porous medium equation does not in general preserve $α$-concavity of the pressure for $0\leα<1/2$ or $1/2<α\le 1$. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $α>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp.
Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\leα<1/2$, we give another set of examples to show that concavity can be broken at a boundary point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_03063 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Non-preservation of $α$-concavity for the porous medium equation Chau, Albert Weinkove, Ben Analysis of PDEs We show that the porous medium equation does not in general preserve $α$-concavity of the pressure for $0\leα<1/2$ or $1/2<α\le 1$. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $α>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp. Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\leα<1/2$, we give another set of examples to show that concavity can be broken at a boundary point. |
| title | Non-preservation of $α$-concavity for the porous medium equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2011.03063 |