Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Yao, Yao

doi:10.1016/j.anihpc.2013.02.002

Cited by 19 publications

(18 citation statements)

References 29 publications

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“…Calvez, Carrillo, and Hoffmann extended the latter result to general power-law interaction potentials −d < k < 0 [41,42]. Finally, in the Newtonian case, Yao showed that every radial solution with compact support will converge to some stationary solution in this family [146], though the asymptotic behavior of non-radial solutions remains unclear.…”

Section: Fair Competition Regime M = M Cmentioning

confidence: 90%

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Carrillo

Craig

Yao

2019

Modeling and Simulation in Science, Engineering and Technology

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Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, and localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method -the blob method for diffusion -to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.Aggregation-diffusion equations: dynamics, asymptotics, and singular limits 5 2 Well-posedness, steady states, and dynamicsWe now give a brief overview of analytical results for aggregation diffusion equations (4), describing conditions that ensure well-posedness or finite-time blow-up of solutions , existence or non-existence of steady states, and long time behavior of solutions. We begin in section 2.1 by considering the classical two-dimensional Keller-Segel equation, where the interaction kernel in equation (4) is given by the Newtonian potential, W (x) = 1 2π ln |x|. The analysis in this particular case will serve as a guideline to understand the equation's behavior for more general interaction kernels. In section 2.2, we review results classifying well-posedness and finite blow-up for general interaction kernels. Finally, in section 2.3, we discuss the steady states and dynamics of solutions in the diffusion-dominated regime. ρ λ (x) := λ 2 ρ(λ x), λ 1.Then, E[ρ λ ] and E[ρ] are related in the following way:

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Section: Fair Competition Regime M = M Cmentioning

confidence: 90%

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Carrillo

Craig

Yao

2019

Modeling and Simulation in Science, Engineering and Technology

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“…Both the aggregation-diffusion equation and constrained aggregation equation have gradient flow structures with respect to the 2-Wasserstein metric. The aggregation-diffusion equation is formally the gradient flow of the sum of an interaction energy and Rényi entropy Over the past fifteen years, there has been significant work on aggregation-diffusion equations, analyzing dynamics of solutions, asymptotic behavior, and minimizers of the energy E m [8,9,12,17,19,22,26,28,29,31,34,37,49,70,71]. The vast majority of the literature has considered one of two choices of interaction potential: either purely attractive power-laws or repulsive-attractive power-laws, K(x) = |x| p /p or K(x) = |x| q /q − |x| p /p for 2 − d p < q 2, q > 0, (1.3) with the convention that |x| 0 /0 = log(|x|).…”

Section: Introductionmentioning

confidence: 99%

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

Craig

Topaloglu

2020

Ann. Inst. H. Poincaré 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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“…Note that Theorem 6 does not imply such rigid behavior at blow-up for solutions with larger than critical mass. Mass comparison methods similar to those employed to prove finite time blow-up and the arguments in [5,45,38,30] are used to show that this concentration cannot occur in finite or infinite time. Hence, the solution must be global and uniformly bounded.…”

Section: Introductionmentioning

confidence: 99%

Global Existence and Finite Time Blow-Up for Critical Patlak--Keller--Segel Models with Inhomogeneous Diffusion

Bedrossian¹,

Kim²

2013

SIAM J. Math. Anal.

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The L 1 -critical parabolic-elliptic Patlak-Keller-Segel system is a classical model of chemotactic aggregation in micro-organisms well-known to have critical mass phenomena [11,9]. In this paper we study this critical mass phenomenon in the context of Patlak-Keller-Segel models with spatially varying diffusivity of the chemo-attractant. The primary issue is how, if possible, one localizes the presence of the inhomogeneity in the nonlocal term. We also provide new blow-up results for critical homogeneous problems with nonlinear diffusion, showing that there exist blow-up solutions with arbitrarily large (positive) initial free energy. For several non-trivial technical reasons, which we discuss in more detail below, we work in dimensions three and higher where L 1 -critical variants of the PKS have porous media-type nonlinear diffusion on the organism density, resulting in finite speed of propagation and simplified functional inequalities.

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Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Cited by 19 publications

References 29 publications

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

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

Global Existence and Finite Time Blow-Up for Critical Patlak--Keller--Segel Models with Inhomogeneous Diffusion

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