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arXiv · 2512.03185

Nonlinear diffusion limit of non-local interactions on a sphere

Mark A. Peletier·Anna Shalova
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Abstract

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.

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Mark A. Peletier, Anna Shalova. 2025-12-02. Nonlinear diffusion limit of non-local interactions on a sphere. https://arxiv.org/abs/2512.03185

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