dorsal/arxiv
View SchemaRenormalization study of two-dimensional convergent solutions of the porous medium equation
| Authors | S. I. Betelu, D. G. Aronson, S. B. Angenent |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9908006 |
| URL | https://arxiv.org/abs/patt-sol/9908006 |
| DOI | 10.1016/S0167-2789(99)00209-2 |
Abstract
In the focusing problem we study a solution of the porous medium equation $u_t=\Delta (u^m)$ whose initial distribution is positive in the exterior of a closed non-circular two dimensional region, and zero inside. We implement a numerical scheme that renormalizes the solution each time that the average size of the empty region reduces by a half. The initial condition is a function with circular level sets distorted with a small sinusoidal perturbation of wave number $k\geq 3$. We find that for nonlinearity exponents m smaller than a critical value which depends on k, the solution tends to a self-similar regime, characterized by rounded polygonal interfaces and similarity exponents that depend on m and on the discrete rotational symmetry number k. For m greater than the critical value, the final form of the interface is circular.
{
"annotation_id": "dcd5da36-3a39-4bf9-b370-c05e3e79bd60",
"date_created": "2026-03-02T18:00:28.876000Z",
"date_modified": "2026-03-02T18:00:28.876000Z",
"file_hash": "642b27bb08adbe9f2d9aa3891357fc1ca7d38c1235c4d542fd533946b32d170a",
"private": false,
"record": {
"abstract": "In the focusing problem we study a solution of the porous medium equation\n$u_t=\\Delta (u^m)$ whose initial distribution is positive in the exterior of a\nclosed non-circular two dimensional region, and zero inside. We implement a\nnumerical scheme that renormalizes the solution each time that the average size\nof the empty region reduces by a half. The initial condition is a function with\ncircular level sets distorted with a small sinusoidal perturbation of wave\nnumber $k\\geq 3$. We find that for nonlinearity exponents m smaller than a\ncritical value which depends on k, the solution tends to a self-similar regime,\ncharacterized by rounded polygonal interfaces and similarity exponents that\ndepend on m and on the discrete rotational symmetry number k. For m greater\nthan the critical value, the final form of the interface is circular.",
"arxiv_id": "patt-sol/9908006",
"authors": [
"S. I. Betelu",
"D. G. Aronson",
"S. B. Angenent"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1016/S0167-2789(99)00209-2",
"title": "Renormalization study of two-dimensional convergent solutions of the porous medium equation",
"url": "https://arxiv.org/abs/patt-sol/9908006"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "36d1c440-b812-443e-9c54-44a131fe6655",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}