Regularized vortex approximation for 2D Euler equations with transport noise

Coghi, Michele; Maurelli, Mario

doi:10.1142/s021949372040002x

Cited by 6 publications

(5 citation statements)

References 79 publications

(71 reference statements)

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“…But there has been little work studying the mean-field problem for the stochastic system (1.1). To the best of our knowledge, the only result is by Coghi and Maurelli [9], which shows that mean-field convergence holds in the periodic case if the Biot-Savart kernel is truncated to length scales much larger than the typical inter-vortex distance N −1/2 . Thus, showing mean-field convergence for the system (1.1), and more generally systems of the form (1.1) with possibly more singular Biot-Savart kernels, is an open problem.…”

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confidence: 91%

“…Remark 1.4. We have chosen to work on R 2 , and not on T 2 as in the prior works [14,6,9] on the SPVM and stochastic Euler equation, in order to parallel our prior work [39] on the deterministic mean-field problem and because the interaction potential g is explicit on R 2 . We expect that one can adapt our proof to the periodic setting, in particular using transference results (e.g., see [18,Section 4.3]) to carry over the singular integral estimates discussed in Appendix A.…”

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confidence: 99%

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The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise

Rosenzweig

2020

Preprint

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In [14], Flandoli, Gubinelli, and Priola proposed a stochastic variant of the classical point vortex system of Helmholtz [20] and Kirchoff [24] in which multiplicative noise of transport-type is added to the dynamics. An open problem in the years since is to show that in the mean-field scaling regime, in which the circulations are inversely proportional to the number of vortices, the empirical measure of the system converges to a solution of a two-dimensional Euler vorticity equation with multiplicative noise. By developing a stochastic extension of the modulated-energy method of Serfaty [45,46] and Duerinckx [13] for mean-field limits of deterministic particle systems and by building on ideas introduced by the author [39,40] for studying such limits at the scaling-critical regularity of the mean-field equation, we solve this problem under minimal assumptions.

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confidence: 91%

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confidence: 99%

The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise

Rosenzweig

2020

Preprint

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“…This consists in studying the mean behaviour of the point-vortex system (with or without perturbations) as their number N → +∞ while their individual circulation scales like 1/N. For more details about mean-field limits for the Euler point-vortex and generalisations, see [5,9,27,39,43,44,47].…”

Section: Introductionmentioning

confidence: 99%

Hölder regularity for collapses of point-vortices

Donati,

Godard-Cadillac

2023

Nonlinearity

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The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. In these models the kernel of the Biot–Savart law is a power function of exponent − α . It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to the time of collapse. The Hölder exponent obtained is 1 / ( α + 1 ) and this exponent is proved to be optimal for all α by exhibiting an example of a three-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as t → T, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established: we prove that their distance to the boundary is Hölder regular.

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“…One is the convergence of the empirical measure to a stochastic 2D Euler equation, where stochasticity reflects the random environment, still present in the limit. This has been done in [4] under Lipschitz condition on the interaction kernel; see [5] for a scaling limit result on point vortices with regularized Biot-Savart kernel and suitably chosen regularizing parameter, and the recent preprint [23] for a mean field limit without smoothing the Biot-Savart kernel. Another face, the one considered here, is to rescale the space covariance of the noise, simultaneously with the increasing number of particles, in such a way that the noise becomes more and more uncorrelated, going heuristically in the direction of the independent additive noises acting on different particles.…”

Section: Introductionmentioning

confidence: 99%