An integro-differential equation arising as a limit of individual cell-based models

Bodnar, Marek; Velázquez, Juan J. L.

doi:10.1016/j.jde.2005.07.025

Cited by 113 publications

(129 citation statements)

References 16 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 one dimension, in the inviscid case (i.e. ν = 0) and for general choices of the kernel K, equation (1.1) has been considered by Bodnar and Velázquez [4]. There by an ODE argument the authors proved the local well-posedness of (1.1) without the diffusion term for C 1 initial data.…”

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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“…It can be shown that, for suitable choices of V (x), there are many non homogeneous steady states of the Eq. (4.5) for potentials containing an attractive and a repulsive part (see [7]). In some special cases we can give a detailed description of some steady states.…”

Section: Derivation Of the Equations: Potentials With An Attractive Partmentioning

confidence: 99%

Derivation of macroscopic equations for individual cell‐based models: a formal approach

Bodnar

Velázquez

2005

Math Methods in App Sciences

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100

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-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 by means of elementary arguments several macroscopic equations that describe the evolution of cell density. Some new results are also obtained -a formal derivation of a limit equation in the case of attractive potential as well as in the case of repulsive potential with a hard-core part are presented Finally, we discuss the possible relevance of those results within the framework of individual cell-based models. Several classes of potentials, including hard-core, repulsive and potentials with attractive parts are discussed. The effect of noise terms in the equation is also considered.

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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%

“…Recently, the finite time blow up problem of (1) has drawn much attention. The existence and uniqueness of solutions for rough initial data and singular potential K has been proven for both one dimension 2,4 and n space dimensions. 20 Finite-time blow-up of solutions under rotationally symmetric kernels with a Lipschitz point at the origin is also known.…”

Section: Introductionmentioning

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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An integro-differential equation arising as a limit of individual cell-based models

Cited by 113 publications

References 16 publications

Wellposedness and regularity of solutions of an aggregation equation

Wellposedness and regularity of solutions of an aggregation equation

Derivation of macroscopic equations for individual cell‐based models: a formal approach

Stability and clustering of self-similar solutions of aggregation equations

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