Swarm dynamics and equilibria for a nonlocal aggregation model

By Razvan C. Fetecau, Yanghong Huang, Theodore Kolokolnikov. 2011

185 citations recorded by the source corpus.

We consider the aggregation equation ρt − ∇ · (ρ∇K * ρ) = 0 in , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which is the steady-state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.

Subfields: Computer Networks and Communications, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/d68e4db8c709d7c50583def2b269ddf5ed04af86

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Citations in the corpus

Oscillators that sync and swarm (2017, methodology): “In Supplementary Note 1 we uset a technique used by Kololnikov et al [50] when studying swarms to derive the following pair of integral -1.5 -1.0 -0.5 0.5 1.0 1.5x -1.5 -1.0 -0.5 0.5 1.0 1.5 y(a) -1.5 -1.0 -0.5 0.5 1.0 1.5x -1.5 -1.0 -0.5 0.5 1.0 1.5 y(b) ⋯ -1.5 -1.0 -0.5 0.5 1.0 1.5x -1.5 -1.0 -0.5”
Ring patterns and their bifurcations in a nonlocal model of biological swarms (2015, background): “Introduction Mathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10, 6, 2, 26, 25, 14, 7, 13, 27, 19, 33, 32, 23, 11, 17, 37, 38, 34, 3”
Dimensionality of Local Minimizers of the Interaction Energy (2012, background): “Recent works [23, 36, 19, 35, 22, 32, 30, 17, 18, 3] have shown that such repulsive-attractive potentials lead to the emergence of surprisingly rich 1 ar X iv :1 21 0.”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, background): “2) These functionals are directly connected to a class of self-assembly/aggregation models which recently have received much attention (see for example, [9, 11, 12, 19, 33, 36, 38, 39, 40, 47, 54, 58, 59]).”
Stationary states of quadratic diffusion equations with long-range attraction (2011, result): “After a first version of the present paper, [30] studied a very special repulsive-attractive equation without diffusion and proved a uniqueness result using a similar method.”