Confinement for repulsive-attractive kernels

Balagué, D.; Carrillo, José A.; Yao, Yao

doi:10.3934/dcdsb.2014.19.1227

Cited by 28 publications

(31 citation statements)

References 40 publications

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“…For the repulsive-attractive case, the requirement p < q ensures that the nonlocal interactions are repulsive at short length scales and attractive at long length scales. This competition between short-range and long-range effects leads to rich pattern formation in both the steady states of solutions and the minimizers of the corresponding energy E m ; see [6,7,13,33,35,40,43,45,65].…”

Section: Introductionmentioning

confidence: 99%

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

Craig

Topaloglu

2020

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.

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Section: Introductionmentioning

confidence: 99%

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

Craig

Topaloglu

2020

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…In many biological applications the number of interacting particles is large and one may consider the underlying continuum formulation of (1.1), which is known as the aggregation equation [9,11,42] and of the form ρ t + ∇ · (ρu) = 0, u = −∇W * ρ, (1.2) where u = u(t, x) is the macroscopic velocity field and ρ = ρ(t, x) denotes the density of particles at location x ∈ R n at time t > 0. The aggregation equation (1.2) has been studied extensively recently, mainly in terms of its gradient flow structure [2,29,30,44,54], the blow-up dynamics for fully attractive potentials [9,10,21,28], and the rich variety of steady states for repulsiveattractive potentials [3,4,5,8,10,19,20,22,23,38,39,50,55,56,27,26,31]. In biological applications, the interactions determined by the force F , or equivalently the interaction potential W , are usually described by short-range repulsion, preventing collisions between the individuals, as well as long-range attraction, keeping the swarm cohesive [46,47].…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Line Patterns of an Anisotropic Interaction Model

Carrillo¹,

Düring²,

Kreußer

et al. 2019

SIAM J. Appl. Dyn. Syst.

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Motivated by the formation of fingerprint patterns we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. We show that there exists a preferred direction of straight lines, i.e. straight vertical lines can be stable for sufficiently many particles, while other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles and relate these results to the Kücken-Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.AMSC: 35B36, 35Q92, 70F10, 70F45, 82C22

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“…pW˚ρq pxqρpdxq and the continuum equation corresponding to (1.2), also referred to as the aggregation equation [8,50,13,54], reads ρ t`∇¨p ρuq " 0, u "´∇W˚ρ (1.3) where u " upt, xq is the macroscopic velocity field. The aggregation equation (1.3) whose wellposedness has been proved in [12] has extensively been studied recently, mainly in terms of its gradient flow structure [55,34,65,2,35], the blow-up dynamics for fully attractive potentials [8,25,11,32], and the rich variety of steady states [43,44,7,62,27,26,11,4,3,66,67,5,21,24,27]. If the radially symmetric potential W is purely attractive, e.g.…”

Section: Introductionmentioning

confidence: 99%

Pattern formation of a nonlocal, anisotropic interaction model

Burger

Düring

Kreußer

et al. 2018

Math. Models Methods Appl. Sci.

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Abstract. We consider a class of interacting particle models with anisotropic, repulsiveattractive interaction forces whose orientations depend on an underlying tensor field. An example of this class of models is the so-called Kücken-Champod model describing the formation of fingerprint patterns. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the equilibria of the corresponding mean-field partial differential equation and investigate pattern formation numerically in two dimensions by studying the dependence of the parameters in the model on the resulting patterns.

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Confinement for repulsive-attractive kernels

Cited by 28 publications

References 40 publications

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

Stability Analysis of Line Patterns of an Anisotropic Interaction Model

Pattern formation of a nonlocal, anisotropic interaction model

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