Existence of ground states for aggregation-diffusion equations

Carrillo, José A.; Delgadino, Matías G.; Patacchini, Francesco S.

doi:10.1142/s0219530518500276

Cited by 25 publications

(50 citation statements)

References 58 publications

(98 reference statements)

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“…Section 6 is devoted to the existence of global minimizers in light of the phenomena discussed in Section 5. Theorem 6.1 establishes a sharp condition for existence in domains of type D = F × R d−m with an effective volume dimension of d − m, where F ⊂ R m is a compact m-dimensional set with 0 < m < d. This result serves as a generalization of the sharp existence condition in free space established in [13]. Theorems 6.3 and 6.4 then provide sufficient conditions for existence in general domains with boundaries, which involve establishing a minimal set of requirements on V for confinement, in light of the metastable translations discussed in Section 5.2 that occur when V = 0.…”

Section: Introductionmentioning

confidence: 77%

“…In the presence of diffusion (ν > 0), the focus is the competition between local repulsion effects (diffusion) and nonlocal attractive interactions that provides existence (or lack of) energy minimizers. This delicate balance of such forces was recently investigated by Carillo et al in [13] for the case of free space; this work is central to our paper and we will return to it frequently throughout.…”

Section: Introductionmentioning

confidence: 84%

“…Extrema of E ν are defined as in [13]: for r > 0 we define a W p -r local minimizer of E ν to be a measure ρ such that…”

Section: )mentioning

confidence: 99%

“…First, we highlight in Theorem 3.1 the property that minimizers of E ν for every ν > 0 are supported on the whole domain D. This is briefly mentioned in [13] for free space and is justified by the authors using [Minimizers have full support] Let Assumptions 1 and 2 hold. In addition, assume that there exist constants p K ≥ 0, R K > 0, and C K > 0 such that…”

Section: Critical Points Of the Energymentioning

confidence: 99%

“…Notation and assumptions are introduced in Section 2. In Section 3 we establish some useful properties of critical points of the energy E ν , including a new energy argument showing that minimizers ρ satisfy supp (ρ) = D. Section 4 contains a brief review of various results in free space established in [13], while Sections 5 and 6 present our results on energy minimization in domains with boundaries.…”

Section: Introductionmentioning

confidence: 99%

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Equilibria of an aggregation model with linear diffusion in domains with boundaries

Messenger

Fetecau

2020

Math. Models Methods Appl. Sci.

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We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends explicitly on a certain volume filling property of the domain, and (ii) metastable mass translation occurs in domains without sufficient symmetry. From the first property, we present a sharp condition for existence (resp. non-existence) of global minimizers in a large class of domains, analogous to results in free space, and from the second property, we identify that external forces are necessary to confine the swarm and grant existence of global minimizers in general domains. We also introduce a numerical method for computing critical points of the energy and give examples to motivate further research.

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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 84%

“…Extrema of E ν are defined as in [13]: for r > 0 we define a W p -r local minimizer of E ν to be a measure ρ such that…”

Section: )mentioning

confidence: 99%

Section: Critical Points Of the Energymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equilibria of an aggregation model with linear diffusion in domains with boundaries

Messenger

Fetecau

2020

Math. Models Methods Appl. Sci.

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Uniqueness and Nonuniqueness of Steady States of Aggregation‐Diffusion Equations

Delgadino

Yan

Yao

2020

Comm Pure Appl Math

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We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean‐field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/nonuniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being m = 2. In the case m ≥ 2, we show that for any attractive potential the steady state is unique for a fixed mass. In the case 1 < m < 2, we construct examples of smooth attractive potentials such that there are infinitely many radially decreasing steady states of the same mass. For the uniqueness proof, we develop a novel interpolation curve between two radially decreasing densities, and the key step is to show that the interaction energy is convex along this curve for any attractive interaction potential, which is of independent interest. © 2020 Wiley Periodicals LLC.

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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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Existence of ground states for aggregation-diffusion equations

Cited by 25 publications

References 58 publications

Equilibria of an aggregation model with linear diffusion in domains with boundaries

Equilibria of an aggregation model with linear diffusion in domains with boundaries

Uniqueness and Nonuniqueness of Steady States of Aggregation‐Diffusion Equations

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

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