Nonlinear diffusion limit of non-local interactions on a sphere

2025/12/02 by Mark A. Peletier, Peletier, Mark A., Anna Shalova +1 · 3 citations
Mathematics · Computer Science · #Mathematical Biology Tumor Growth #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.03185

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.

Citations

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Cited by

• Ground states and phase transitions for an aggregation model with fast diffusion on sphere
• A deterministic particle approximation for a fourth-order equation
• Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime

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• Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
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• The aggregation-diffusion equation with energy critical exponent
• Aggregation-diffusion phenomena: from microscopic models to free boundary problems