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

Craig, Katy; Topaloglu, Ihsan

doi:10.1016/j.anihpc.2019.10.003

Cited by 20 publications

(18 citation statements)

References 65 publications

(104 reference statements)

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“…Finally, provided that sufficient a priori estimates hold along the flow, gradient flows of the regularized energies converge to the solution of the aggregation diffusion equation with initial data ρ 0 as ε → 0 and d 2 (ρ N 0 , ρ 0 ) → 0. More recently, Craig and Topaloglu have demonstrates numerically that this method also provides a robust approach for simulating aggregation diffusion equations, by sending the diffusion exponent m → +∞ as the regularization and the discretization of the initial data are refined [82].…”

Section: Methodsmentioning

confidence: 99%

“…We begin in section 5.1 by describing the details of our numerical implementation, which include various refinements over previous works, such as regridding to reduce the number of particles required for convergence [55,82]. In section 5.2, we provide several numerical examples of the slow diffusion limit and properties of the constrained aggregation equation, particularly critical mass behavior relating to open problems in geometric shape optimization [34,82,91]. In section 5.3, we give numerical examples illustrating the relationship between singular limits and metastability behavior, both as aggregation becomes localized and as diffusion vanishes.…”

Section: Simulations Via the Blob Methods For Diffusionmentioning

confidence: 99%

“…On the other hand, if the interaction potential is a repulsive-attractive power-law potential (2) for A ≥ 2 − d, Carrillo and Wang prove a global-in-time L ∞ bound for all m ≥ 1, under the assumption that solutions exist locally in time [70]. General well-posedness results were recently obtained by Craig and Topaloglu, without any assumptions on the specific structure of the attractive and repulsive parts of the interaction potential [82]. In particular, under weak regularity assumptions on the interaction potential, which roughly correspond to being no more singular than the Newtonian potential, they prove that solutions of aggregation diffusion equations exist and provide sufficient conditions for global in time uniqueness.…”

Section: General Interaction Potentialsmentioning

confidence: 99%

“…More recently, Craig and Topaloglu [82] proved that minimizers and gradient flows of E m do converge to minimizers and gradient flows of E ∞ in the m → ∞ limit for a general class of interaction potentials W , including both attractive and repulsive-attractive power-law potentials with singularity up to and including the Newtonian potential. This work was largely inspired by the following related question in geometric shape optimization: can we characterize set-valued or patch minimizers E ∞ , i.e.…”

Section: Steady States and Dynamicsmentioning

confidence: 99%

“…In the fluids community, it is well known that this relationship (25) between the interparticle distance and the regularization needs to hold globally in time for the numerical simulation to agree well with exact solutions. In previous work on aggregation and aggregation diffusion equations [55,79,82], this was accomplished by taking the spatial discretization h very small. In the present work, we accomplish this by using the formula for the approximate density (24) to re-initialize our particle approximation (20) whenever the maximum interparticle distance exceeds 1.5h.…”

Section: Numerical Implementationmentioning

confidence: 99%

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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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Section: Methodsmentioning

confidence: 99%

Section: Simulations Via the Blob Methods For Diffusionmentioning

confidence: 99%

Section: General Interaction Potentialsmentioning

confidence: 99%

Section: Steady States and Dynamicsmentioning

confidence: 99%

Section: Numerical Implementationmentioning

confidence: 99%

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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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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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On the incompressible limit for a tumour growth model incorporating convective effects

David,

Schmidtchen

2023

Comm Pure Appl Math

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In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self‐propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density‐based model and a geometry free‐boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

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

Cited by 20 publications

References 65 publications

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

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

Uniqueness and Nonuniqueness of Steady States of Aggregation‐Diffusion Equations

On the incompressible limit for a tumour growth model incorporating convective effects

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