×

Self-organization on Riemannian manifolds. (English) Zbl 1431.35218

Summary: 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 and an application of the model to aggregation on the rotation group \(\mathrm{SO}(3)\) are also presented.

MSC:

35R01 PDEs on manifolds
34C40 Ordinary differential equations and systems on manifolds
35B36 Pattern formations in context of PDEs
35Q70 PDEs in connection with mechanics of particles and systems of particles
PDFBibTeX XMLCite
Full Text: DOI arXiv