Mean-Field Dynamics for Ginzburg–Landau Vortices with Pinning and Forcing

Duerinckx, Mitia; Serfaty, Sylvia

doi:10.1007/s40818-018-0053-0

Cited by 6 publications

(5 citation statements)

References 83 publications

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“…The lake equations (1.1) can be derived formally from the three-dimensional Euler equations [8] and have been justified mathematically in the periodic case [40]. They appear in the mean-field limit for the Gross-Pitaevskii equation, which is the Schrödinger flow for the Ginzburg-Landau energy, under forcing and pinning [18]. Weak solutions of the Cauchy problem for the lake equations (1.1) exist globally [23,30,32,38,39]; these solutions are unique [5,30,38] and as smooth as the data permits it ( [23,31,39] and appendix B below).…”

Section: Introductionmentioning

confidence: 99%

Vortex Motion for the Lake Equations

Dekeyser

Schaftingen

2020

Commun. Math. Phys.

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The lake equationsmodel the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth b : D → [0, +∞) is small in comparison with the size of its twodimensional projection D ⊂ R 2 . When the depth b is positive everywhere in D and constant on the boundary, we prove that the vorticity of solutions of the lake equations whose initial vorticity concentrates at an interior point is asympotically a multiple of a Dirac mass whose motion is governed by the depth function b.

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Section: Introductionmentioning

confidence: 99%

Vortex Motion for the Lake Equations

Dekeyser

Schaftingen

2020

Commun. Math. Phys.

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“…The relative entropy method, initiated in [89] in the context of hydrodynamics of Ginzburg-Landau and now has been extensively used for hydrodynamics limits (see chapter 6 in [59]), is maybe the closest to the approach developed here. A similar approach, namely a modulated energy argument, was introduced in [82] to investigate mean field limits for quantum vortices (see also [26]), and has been used in [25] for gradient flows with Riesz-like potentials and in [83] for 1st order Coulomb flows. We also refer to [34] for a different, trajectorial, view on the role of the entropy in SDEs.…”

Section: Main Results For Non-vanishing Diffusionmentioning

confidence: 99%

“…In particular, [26,82,83] recently introduced a relative entropy method at the the level of the empirical measure based on the energy of the system. This allows to obtain quantitative estimates, in particular for deterministic settings, with quite singular interactions of Riesz potential form.…”

Section: Discussionmentioning

confidence: 99%

“…The method performs especially well on gradient flows (where our present techniques are sub-optimal) and allows to obtain the mean field limit for general Riesz potentials (including Coulomb in 2D). This approach can also be used when the discrete dynamics is not immediately under the form of an aggregation equation, with Ginzburg-Landau vortices in [26]. The technique also allowed to include Coulomb interaction in any dimension in [83].…”

Section: As We Remarked Abovementioning

confidence: 99%

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Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels

Jabin,

Wang

2017

Preprint

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We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our result only requires very weak regularity on the interaction kernel in the negative Sobolev space Ẇ −1,∞ , thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes.

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“…In [18] Duerinckx gave an other proof of the mean-field limit of several Riesz interaction gradient flows using a "modulated energy" that was introduced by Serfaty in [47]. Together they also gave a mean-field limit of Ginzburg-Landau vortices with pinning and forcing effects in [19].…”

Section: Introductionmentioning

confidence: 99%

Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects

Ménard¹

2022

Preprint

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In this paper we derive a two dimensional spray model with gyroscopic effects as the mean-field limit of a system modeling the interaction between an incompressible fluid and a finite number of solid particles. This spray model has been studied by Moussa and Sueur (Asymptotic Anal., 2013), in particular the mean-field limit was established in the case of W 1,∞ interactions. First we prove the local in time existence and uniqueness of strong solutions of a monokinetic version of the model with a fixed point method. Then we adapt the proof of Duerinckx and Serfaty (Duke Math. J., 2020) to establish the mean-field limit to the spray model in the monokinetic regime with Coulomb interactions.

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Mean-Field Dynamics for Ginzburg–Landau Vortices with Pinning and Forcing

Cited by 6 publications

References 83 publications

Vortex Motion for the Lake Equations

Vortex Motion for the Lake Equations

Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels

Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects

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