Nonlinear diffusion limit of non-local interactions on a sphere
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917121057882112 |
|---|---|
| author | Peletier, Mark A. Shalova, Anna |
| author_facet | Peletier, Mark A. Shalova, Anna |
| contents | We study an aggregation PDE with competing attractive and repulsive forces on a sphere of arbitrary dimension. In particular, we consider the limit of strongly localized repulsion with a constant attraction term. We prove convergence of solutions of such a system to solutions of the aggregation-diffusion equation with a porous-medium-type diffusion term. The proof combines variational techniques with elements of harmonic analysis on a sphere. In particular, we characterize the square root of the convolution operator in terms of the spherical harmonics, which allows us to overcome difficulties arising due to the convolution on a sphere being non-commutative. The study is motivated by the toy model of transformers introduced by Geshkovski et al. (2025); and we discuss the applicability of the results to this model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear diffusion limit of non-local interactions on a sphere Peletier, Mark A. Shalova, Anna Analysis of PDEs Mathematical Physics We study an aggregation PDE with competing attractive and repulsive forces on a sphere of arbitrary dimension. In particular, we consider the limit of strongly localized repulsion with a constant attraction term. We prove convergence of solutions of such a system to solutions of the aggregation-diffusion equation with a porous-medium-type diffusion term. The proof combines variational techniques with elements of harmonic analysis on a sphere. In particular, we characterize the square root of the convolution operator in terms of the spherical harmonics, which allows us to overcome difficulties arising due to the convolution on a sphere being non-commutative. The study is motivated by the toy model of transformers introduced by Geshkovski et al. (2025); and we discuss the applicability of the results to this model. |
| title | Nonlinear diffusion limit of non-local interactions on a sphere |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2512.03185 |