Propagation of chaos for the 2D viscous vortex model

Fournier, Nicolas; Hauray, Maxime; Mischler, Stéphane

doi:10.4171/jems/465

Cited by 108 publications

(158 citation statements)

References 49 publications

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“…We use the same methods for a subcritical Keller-Segel equation. The proofs are thus sometimes very similar to those in [8] but there are some differences due to the facts that (i) there are no circulation parameter (M N i in [8]): this simplify the situation since we thus deal with solutions which are probabilities, (ii) α = 1 so when we use Hardy-Littlewood-Sobolev's inequality an extra change of variables for the time variable is needed (see Step 1 in the proof of Theorem 1.5 in Section 6) and (iii) the kernel is not the same: it is not divergence-free and we thus have to deal with some additional terms in our computations (see the comments before Proposition 3.1 and in the proof of Theorem 1.5). We can also notice that due to this fact, we have no already known result for the existence and uniqueness of the particle system that we consider.…”

Section: Commentssupporting

confidence: 56%

“…This paper is some kind of adaptation of the work of Fournier, Hauray and Mischler in [8] where they show the propagation of chaos of some particle system for the 2D viscous vortex model. We use the same methods for a subcritical Keller-Segel equation.…”

Section: Commentsmentioning

confidence: 99%

“…In this section, we recall some lemmas stated in [8] and [11] and we state a result on the regularity of the kernel K defined in (1.2). The first result tells us that pairs of particles which law have finite Fisher information cannot be too close.…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Propagation of chaos for a subcritical Keller–Segel model

Godinho¹,

Quiñinao²

2015

Ann. Inst. H. Poincaré Probab. Statist.

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This paper deals with a subcritical Keller-Segel equation. Starting from the stochastic particle system associated with it, we show well-posedness results and the propagation of chaos property. More precisely, we show that the empirical measure of the system tends towards the unique solution of the limit equation as the number of particles goes to infinity.Résumé. Cet article traite de l'équation de Keller-Segel dans un cadre sous-critique. À l'aide du système de particules en lien avec cette équation, nous montrons des résultats d'existence et d'unicité, puis la propagation du chaos pour ce dernier. Plus précisément, nous montrons que la mesure empirique du système tend vers l'unique solution de l'équation limite lorsque le nombre de particules tend vers l'infini. MSC: 65C35where f : R + ×R 2 → R and χ > 0. The force field kernel K : R 2 → R 2 comes from an attractive potential : R 2 → R and is defined by, α ∈ (0, 1).(1.2)The standard Keller-Segel equation correspond to the critical case K(x) = x/|x| 2 (i.e., more singular at x = 0) and it describes a model of chemotaxis, i.e., the movement of cells (usually bacteria or amoebae) which are attracted 966 D. Godinho and C. Quiñinao by some chemical substance that they produce. This equation has been first introduced by Keller and Segel in [15,16]. Blanchet, Dolbeault and Perthame showed in [4] some nice results on existence of global weak solutions if the nonnegative parameter χ (which is the sensitivity of the bacteria to the chemo-attractant) is smaller than 8π/M where M is the initial mass (here M will always be 1 since we will deal with probability measures). For more details on the subject, see [12,13].

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Section: Commentssupporting

confidence: 56%

Section: Commentsmentioning

confidence: 99%

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Propagation of chaos for a subcritical Keller–Segel model

Godinho¹,

Quiñinao²

2015

Ann. Inst. H. Poincaré Probab. Statist.

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“…He also studied the propagation of chaos for the Navier-Stokes equation with the random vortex method without regularized parameters. In a recent important work of Fournier et al [4], the authors significantly improved Osada's result: (i) They proved the propagation of chaos for the 2D viscous vortex model with any positive viscosity coefficient; (ii) the convergence holds in a strong sense, called entropic.…”

Section: Definitionmentioning

confidence: 82%

Propagation of chaos for large Brownian particle system with Coulomb interaction

Liu

Yang

2016

Res Math Sci

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We investigate a system of N Brownian particles with the Coulomb interaction in any dimension d ≥ 2, and we assume that the initial data are independent and identically distributed with a common density ρ 0 satisfyingWe prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For d = 2, we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When N → ∞, the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson-Nernst-Planck equation of single component.

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“…Our proof follows a strategy introduced in [23] for the 2D viscous vortex model. It is based on a DiPerna-Lions renormalization trick (see [21]) which makes possible to get the optimal regularity of solutions for small time and then to follow the uniqueness argument introduced by Ben-Artzi for the 2D viscous vortex model (see [4,10]).…”

Section: 7)mentioning

confidence: 99%

Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case

Fernández

Mischler

2015

Arch Rational Mech Anal

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Abstract. The present paper deals with the parabolic-elliptic Keller-Segel equation in the plane in the general framework of weak (or "free energy") solutions associated to initial datum with finite mass M , finite second moment and finite entropy. The aim of the paper is threefold:(1) We prove the uniqueness of the "free energy" solution on the maximal interval of existence [0, T * ) with T * = ∞ in the case when M ≤ 8π and T * < ∞ in the case when M > 8π. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical L 4/3 Lebesgue norm similarly as for the 2d vorticity Navier-Stokes equation.(2) We prove immediate smoothing effect and, in the case M < 8π, we prove Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables).(3) In the case M < 8π, we also prove weighted L 4/3 linearized stability of the self-similar profile and then universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.

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Propagation of chaos for the 2D viscous vortex model

Cited by 108 publications

References 49 publications

Propagation of chaos for a subcritical Keller–Segel model

Propagation of chaos for a subcritical Keller–Segel model

Propagation of chaos for large Brownian particle system with Coulomb interaction

Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case

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