Aggregation and Spreading via the Newtonian Potential: The Dynamics of Patch Solutions

Bertozzi, Andrea L.; Laurent, Thomas; Léger, Flavien

doi:10.1142/s0218202511400057

Cited by 97 publications

(134 citation statements)

References 57 publications

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“…The inequality 2|ab| ≤ a 2 + b 2 and an integration by parts then yield 4) so that the result (2.1) follows from (2.2) and a Gronwall argument.…”

Section: Strategy Of the Proofmentioning

confidence: 82%

Mean-Field Limits for Some Riesz Interaction Gradient Flows

Duerinckx¹

2016

SIAM J. Math. Anal.

104

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This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty [28] in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open. IntroductionWe consider the energy of a system of N particles in the Euclidean space R d (d ≥ 1) interacting via (repulsive) Riesz pairwise interactions:where the interaction kernel is given bywith c d,s > 0 some normalization constants. We note that the Coulomb case corresponds to the choice s = d − 2, d ≥ 2. Particle systems with more general Riesz interactions as considered here are extensively motivated in the physics literature (cf. for instance [2,20]), as well as in the context of approximation theory with the study of Fekete points (cf.[14] and the references therein). Recently, a detailed description of such systems beyond the mean-field limit in the static case was obtained in [21], and also in [17] for non-zero temperature. In the present contribution, we are rather interested in the dynamics of such systems, and more precisely in a rigorous justification of the mean-field limit of their gradient flow evolution as the number N of particles tends to infinity, which has indeed remained an open problem wheneverWe thus consider the trajectories x t i,N driven by the corresponding flow, i.e. the solutions to the following system of ODEs:where (xis a sequence of N distinct initial positions. Since energy can only decrease in time and since the interaction is repulsive, particles cannot collide, and moreover it is easily seen that a particle cannot escape to infinity in finite time; from these observations and from the Picard-Lindelöf theorem, we may conclude that the trajectories x t i,N are smooth and well-defined on the whole of R + := [0, ∞). As the number of particles gets large, we would naturally like to pass to a continuum description of the system, in terms of the particle density distribution. For that purpose, we define the empirical measure associated with the point-vortex dynamics:2) 1 and the question is then to understand the limit of µ t N as N ↑ ∞. More precisely, assuming convergence at initial time This equation in the weak sense just means the following:As far as existence issues as well as basic properties of the solutions of (1.3) are concerned, we refer to [9,8] . Schochet's original paper [27] was actually only concerned with the mean-field limit for a particle approximation of the 2D Euler equation, but the same argument directly applies to the present setting. However, due to a possible lack of uniqueness of L 1 weak solutions to equation (1.3), Schochet [27] could only prove that the empirical measure µ t N converges up to a subsequence to some solution of (1.3). The key idea, which only holds for logarithmic i...

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“…The inequality 2|ab| ≤ a 2 + b 2 and an integration by parts then yield 4) so that the result (2.1) follows from (2.2) and a Gronwall argument.…”

Section: Strategy Of the Proofmentioning

confidence: 82%

Mean-Field Limits for Some Riesz Interaction Gradient Flows

Duerinckx¹

2016

SIAM J. Math. Anal.

104

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“…As such, it is related to nonlocal aggregation and swarming models, which have attracted a lot of mathematical interest in recent years (see e.g. [8,14] and the references therein); they consist in replacing the Coulomb potential (−△) −1 by a convolution with a more general kernel corresponding to an attractive (rather than repulsive) nonlocal interaction. Equation (1.4) was first studied by Lin and Zhang [39], who established global existence for vortex-sheet initial data ω| t=0 ∈ P(R 2 ), and uniqueness in some Zygmund space.…”

Section: Relation To Previous Workmentioning

confidence: 99%

Well-posedness for mean-field evolutions arising in superconductivity

Duerinckx¹,

Fischer²

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in [24] as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.

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“…Yet, the good news is the fact that this bound will be completely independent of the choice of f and of χ. Essentially, as the reader can easily guess, we are providing L ∞ bounds that hold for the system without diffusion, as in [33,5], which is compatible with an estimate exploding in time. The interest of our work mainly resides in its discrete-in-time nature, namely the fact that the estimate is proven along the JKO scheme.…”

Section: Introductionmentioning

confidence: 99%

$L^\infty $ estimates for the JKO scheme in parabolic-elliptic Keller-Segel systems

Carrillo

Santambrogio

2017

Quart. Appl. Math.

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Abstract. We prove L ∞ estimates on the densities that are obtained via the JKO scheme for a general form of a parabolic-elliptic Keller-Segel type system, with arbitrary diffusion, arbitrary mass, and in arbitrary dimension. Of course, such an estimate blows up in finite time, a time proportional to the inverse of the initial L ∞ norm. This estimate can be used to prove short-time well-posedness for a number of equations of this form regardless of the mass of the initial data. The time of existence of the constructed solutions coincides with the maximal time of existence of Lagrangian solutions without the diffusive term by characteristic methods.

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Aggregation and Spreading via the Newtonian Potential: The Dynamics of Patch Solutions

Cited by 97 publications

References 57 publications

Mean-Field Limits for Some Riesz Interaction Gradient Flows

Mean-Field Limits for Some Riesz Interaction Gradient Flows

Well-posedness for mean-field evolutions arising in superconductivity

$L^\infty $ estimates for the JKO scheme in parabolic-elliptic Keller-Segel systems

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