On an aggregation model with long and short range interactions

Burger, Martin; Capasso, Vincenzo; Morale, Daniela

doi:10.1016/j.nonrwa.2006.04.002

Cited by 145 publications

(156 citation statements)

References 18 publications

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“…¿From the mathematical point of view aggregation equations have been studied extensively (see e.g. [2], [3], [4], [5], [6], [21], [25] and [36]). In one dimension, in the inviscid case (i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Wellposedness and regularity of solutions of an aggregation equation

Liu¹,

Rodrigo

2010

Rev. Mat. Iberoam.

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We consider an aggregation equation in R d , d ≥ 2 with fractional dissipation: u t + ∇ · (u∇K * u) = −νΛ γ u, where ν ≥ 0, 0 < γ ≤ 2 and K(x) = e −|x| . In the supercritical case, 0 < γ < 1, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, γ = 1, we prove the global wellposedness for initial data having a small L 1 x norm. In the subcritical case, γ > 1, we prove global wellposedness and smoothing of solutions with general L 1 x initial data.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the case of sufficiently small diffusion (a(ρ) = ρ 2 ) they proved the existence of stationary solutions with small support. Burger, Capasso and Morale [6] studied the well-posedness of an equation similar to (1.1) but with a different diffusion term:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Wellposedness and regularity of solutions of an aggregation equation

Liu¹,

Rodrigo

2010

Rev. Mat. Iberoam.

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“…Some continuum models in the literature [41,42,7] include attraction-repulsion mechanisms and spatial diffusion to deal with random effects. Other continuum models are based on hydrodynamic descriptions [13,10] derived from mean-field particle limits.…”

Section: Introductionmentioning

confidence: 99%

Collective Behavior of Animals: Swarming and Complex Patterns

Cañizo¹,

Carrillo²,

Rosado³

2010

Arbor

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RESUMEN: En esta nota repasamos algunos modelos basados en individuos para describir el movimiento colectivo de agentes, a lo ABSTRACT:In this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs (ordinary differential equations) exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE (partial differential equations) models are questions of actual interest in mathematical biology research.

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“…, n 4 }} on the point p 3,1 , with 1, j = 0. We can further decompose 1, j uniquely into a tangential component 1 1, j in the plane determined by three points p 3,1 , p 3,2 , p 3,3 and a tangential component 2 1, j in the plane determined by three points p 3,1 , p 3,3 , p 3,4 . The magnitude of the perturbation 1 1, j satisfies Eq.…”

Section: Stability Of Clusters In General Space Dimensionsmentioning

confidence: 99%

“…The magnitude of the perturbation 1 1, j satisfies Eq. (33) with c determined by (30) and, by Taylor expanding (12), the perturbation 1 1, j has a higher order therefore negligible effect on particles Y 4, j located at the point p 3,4 . Thus any perturbation { 1, j = 1 1, j , i , j = 0, ∀i = 1} with arbitrary 1 1, j and j 1 1, j = 0 is an eigenvector of the linearization of (12) with eigenvalue c .…”

Section: Stability Of Clusters In General Space Dimensionsmentioning

confidence: 99%

Stability and clustering of self-similar solutions of aggregation equations

Sun

Uminsky

Bertozzi

2012

Journal of Mathematical Physics

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In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K * ρ) in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd, d ⩾ 2, where K(r) = rγ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math. 70, 2582–2603 (2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ > 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two-dimensional (in-)stability implies n-dimensional (in-)stability.

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On an aggregation model with long and short range interactions

Cited by 145 publications

References 18 publications

Wellposedness and regularity of solutions of an aggregation equation

Wellposedness and regularity of solutions of an aggregation equation

Collective Behavior of Animals: Swarming and Complex Patterns

Stability and clustering of self-similar solutions of aggregation equations

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