Nonlinear diffusion limit of non-local interactions on a sphere

TL;DR

This paper analyzes the limit behavior of an aggregation PDE with non-local interactions on a sphere, showing convergence to a porous-medium type diffusion equation using harmonic analysis and variational methods.

Contribution

It introduces a novel analysis of the nonlinear diffusion limit on a sphere, characterizing the convolution operator via spherical harmonics and addressing non-commutativity issues.

Findings

Proves convergence of solutions to a porous-medium type equation.

Characterizes the convolution operator using spherical harmonics.

Provides insights relevant to transformer models in machine learning.

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.