Small inertia regularization of an anisotropic aggregation model

@article{Evers2016SmallIR,
title={Small inertia regularization of an anisotropic aggregation model},
author={Joep H. M. Evers and R. Fetecau and W. Sun},
journal={arXiv: Classical Analysis and ODEs},
year={2016}
}
• Published 2016
• Mathematics, Physics
• arXiv: Classical Analysis and ODEs
We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter $\varepsilon$, which can be interpreted as inertia or response time. We examine rigorously the limit $\varepsilon \to 0$ of solutions to the relaxation system. Of major interest is how discontinuous (in velocities) solutions to the first-order model are captured in the zero-inertia limit. We find that near such discontinuities… Expand
9 Citations

Figures from this paper

Pattern formation of a nonlocal, anisotropic interaction model
• Physics, Mathematics
• 2016
We consider a class of interacting particle models with anisotropic, repulsive–attractive interaction forces whose orientations depend on an underlying tensor field. An example of this class ofExpand
Filippov flows and mean-field limits in the kinetic singular Kuramoto model
The agent-based singular Kuramoto model was proposed in [60] as a singular version of the Kuramoto model of coupled oscillators that is consistent with Hebb's rule of neuroscience. In such paper, theExpand
1 A ug 2 02 1 MEAN-FIELD PARTICLE SWARM OPTIMIZATION
In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods.Expand
An anisotropic interaction model for simulating fingerprints
• Physics, Mathematics
• Journal of mathematical biology
• 2019
It is shown that fingerprint patterns can be modeled as stationary solutions by choosing the underlying Tensor field appropriately, and this dependence on the tensor field leads to complex, anisotropic patterns. Expand
An anisotropic interactionmodel for simulating fingerprints
Evidence suggests that both the interaction of so-calledMerkel cells and the epidermal stress distribution play an important role in the formation of fingerprint patterns during pregnancy. TomodelExpand
Anisotropic nonlinear PDE models and dynamical systems in biology
This thesis was supported by the EPSRC, the MSCA-RISE projects CHiPS and NoMADS, the Cambridge Commonwealth, European & International Trust, the German Academic Scholarship Foundation, the CambridgeExpand
Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives
This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of ...
Mean-field particle swarm optimization
• S. Grassi, Hui Huang, Jinniao Qiu
• Computer Science, Mathematics
• ArXiv
• 2021
A continuous formulation via second-order systems of stochastic differential equations of metaheuristic methods based on particle swarm optimization (PSO) for the global minimization of a non-convex and possibly non-smooth high dimensional objective function. Expand
An anisotropic interaction model with collision avoidance
In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general interacting particle system, as used for swarming or follower-leader dynamics. AnExpand

References

SHOWING 1-10 OF 32 REFERENCES
Anisotropic interactions in a first-order aggregation model
• Mathematics
• 2014
We extend a well studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modelled by limited sensorial perception of individuals, thatExpand
First-order aggregation models with alignment
• Mathematics
• 2016
Abstract We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equationExpand
First-order aggregation models and zero inertia limits
• Mathematics
• 2015
Abstract We consider a first-order aggregation model in both discrete and continuum formulations and show rigorously how it can be obtained as zero inertia limits of second-order models. In theExpand
Asymptotic Dynamics of Attractive-Repulsive Swarms
• Mathematics, Computer Science
• SIAM J. Appl. Dyn. Syst.
• 2009
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
Swarm dynamics and equilibria for a nonlocal aggregation model
• Mathematics
• 2011
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 potentialsExpand
Phase transition and diffusion among socially interacting self-propelled agents
• Mathematics, Physics
• 2012
We consider a hydrodynamic model of swarming behavior derived from the kinetic description of a particle system combining a noisy Cucker-Smale consensus force and self-propulsion. In the largeExpand
Derivation of macroscopic equations for individual cell-based models: a formal approach
• Mathematics
• 2005
In this paper we review the theory of cells (particles) that evolve according to a dynamics determined by friction and that interact between themselves by means of suitable potentials. We derive byExpand
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations
• Mathematics
• 2011
In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interactingExpand
Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn
• Mathematics
• 2007
We consider the aggregation equation $$u_t + \nabla \cdot(u \nabla K\,*\,u) = 0$$ in Rn, n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin,Expand
Stability of ring patterns arising from two-dimensional particle interactions.
• Physics, Medicine
• Physical review. E, Statistical, nonlinear, and soft matter physics
• 2011
Weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria including triangular shapes, annuli, and spot patterns with N-fold symmetry. Expand