Self-organization on Riemannian manifolds

Razvan C. Fetecau; Beril Zhang

doi:10.3934/jgm.2019020

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arXiv · 1802.06089

Self-organization on Riemannian manifolds

Razvan C. Fetecau·Beril Zhang

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Abstract

We consider an aggregation model that consists of an active transport equation for the macroscopic population density, where the velocity has a nonlocal functional dependence on the density, modelled via an interaction potential. We set up the model on general Riemannian manifolds and provide a framework for constructing interaction potentials which lead to equilibria that are constant on their supports. We consider such potentials for two specific cases (the two-dimensional sphere and the two-dimensional hyperbolic space) and investigate analytically and numerically the long-time behaviour and equilibrium solutions of the aggregation model on these manifolds. Equilibria obtained numerically with other interaction potentials are also presented.

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BibTeX RIS

Razvan C. Fetecau, Beril Zhang. 2018-02-16. Self-organization on Riemannian manifolds. https://doi.org/10.3934/jgm.2019020

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