Swarm dynamics and equilibria for a nonlocal aggregation model

@article{Fetecau2011SwarmDA,
  title={Swarm dynamics and equilibria for a nonlocal aggregation model},
  author={R. Fetecau and Y. Huang and T. Kolokolnikov},
  journal={Nonlinearity},
  year={2011},
  volume={24},
  pages={2681-2716}
}
We consider the aggregation equation ρt −∇ ·(ρ∇K ∗ ρ) = 0i nR n , 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… Expand

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References

SHOWING 1-10 OF 56 REFERENCES
A non-local model for a swarm
Abstract. This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists ofExpand
Asymptotic Dynamics of Attractive-Repulsive Swarms
TLDR
An analytical upper bound is derived for the finite blow-up time after which the solution forms one or more $\delta$-functions of the conservation equation. Expand
Large time behavior of nonlocal aggregation models with nonlinear diffusion
The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local Expand
A Primer of Swarm Equilibria
TLDR
The model can reproduce a model of locust swarms, which in nature are observed to consist of a concentrated population on the ground separated from an airborne group, and quasi-two-dimensionality of the model plays a critical role. Expand
Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups
TLDR
A continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior, which takes the form of a conservation law in two spatial dimensions. Expand
STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS
In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potentialExpand
A Nonlocal Continuum Model for Biological Aggregation
TLDR
A continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal is constructed, and energy arguments are used to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. Expand
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels !
We consider the multidimensional aggregation equation ut − ∇ (u∇K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). InExpand
Self-Similar Blowup Solutions to an Aggregation Equation in Rn
TLDR
Numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇· (u∇K ∗ u), where the kernel K(x )= |x| shows self-similar behavior in which zero mass concentrates at the core at the blowup time. Expand
Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions
  • J. Velázquez
  • Mathematics, Computer Science
  • SIAM J. Appl. Math.
  • 2004
TLDR
This paper analyzes the precise way in which the regularization introduced in the Keller--Segel system stops the aggregation process and yields the formation of concentration regions. Expand
...
1
2
3
4
5
...