Mean-Field Convergence of Point Vortices to the Incompressible Euler Equation with Vorticity in $$L^\infty $$

Rosenzweig, Matthew

doi:10.1007/s00205-021-01735-3

Cited by 11 publications

(23 citation statements)

References 56 publications

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“…With a bit more work and at the expense of worsening the rate of convergence in N , it is possible to relax the regularity assumption to u ∈ B 1 ∞,1 (T d ) (see (2.1) for definition), which is a scalingcritical space for the well-posedness of the equation. It is an interesting mathematical problem whether one can allow for weak solutions u, in the spirit of the author's prior work [30] (see also [28,29]) on the point vortex approximation in dimension d = 2. We plan to address this question in a separate work.…”

Section: Remark 13mentioning

confidence: 99%

“…In [30] (see also [28,29]), the author complemented Serfaty's smearing procedure with the new idea of tracking the relative size of these additive terms through a series of small, possibly time-dependent parameters. One keeps these parameters unspecified until the conclusion of the proof, where they are chosen to optimize the balance of all the error terms accumulated to estimate expressions of the form Term 2 .…”

Section: Comments On the Proofmentioning

confidence: 99%

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On the rigorous derivation of the incompressible Euler equation from Newton’s second law

Rosenzweig

2023

Lett Math Phys

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A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of N particles interacting in $${\mathbb {T}}^d$$ T d , $$d\ge 2$$ d ≥ 2 , via Newton’s second law through a supercritical mean-field limit. Namely, the coupling constant $$\lambda $$ λ in front of the pair potential, which is Coulombic, scales like $$N^{-\theta }$$ N - θ for some $$\theta \in (0,1)$$ θ ∈ ( 0 , 1 ) , in contrast to the usual mean-field scaling $$\lambda \sim N^{-1}$$ λ ∼ N - 1 . Assuming $$\theta \in (1-\frac{2}{d(d+1)},1)$$ θ ∈ ( 1 - 2 d ( d + 1 ) , 1 ) , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as $$N\rightarrow \infty $$ N → ∞ . Han-Kwan and Iacobelli asked if their range for $$\theta $$ θ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit $$N\rightarrow \infty $$ N → ∞ for $$\theta \in (1-\frac{2}{d},1)$$ θ ∈ ( 1 - 2 d , 1 ) . Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for $$\theta $$ θ . Additionally, we show that for $$\theta \le 1-\frac{2}{d}$$ θ ≤ 1 - 2 d , one cannot, in general, expect convergence in the modulated energy notion of distance.

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Section: Remark 13mentioning

confidence: 99%

Section: Comments On the Proofmentioning

confidence: 99%

On the rigorous derivation of the incompressible Euler equation from Newton’s second law

Rosenzweig

2023

Lett Math Phys

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“…Rosenzweig proved in [44] the mean-field convergence of point vortices without assuming lipschitz regularity for the limit velocity field, using the same energy as in [48] with refined estimates. Remark that it ensures that the point vortices converges to any Yudovich solutions of the Euler equation (see [49]).…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.4. One could replace the compact support assumptions by some logarithmic decrease of the solutions ω and ρ at infinity as done in [18] and [44] but for the sake of simplicity we will only consider solutions with compact support.…”

Section: Introductionmentioning

confidence: 99%

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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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A strong form of propagation of chaos for Cucker–Smale model

Wu,

Wang,

Liu

2024

Z. Angew. Math. Phys.

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Mean-Field Convergence of Point Vortices to the Incompressible Euler Equation with Vorticity in $$L^\infty $$

Abstract: The Version of Record is the version of the article after copy-editing and typesetting, and connected to open research data, open protocols, and open code where available. Any supplementary information can be found on the journal website, connected to the Version of Record.

Cited by 11 publications

References 56 publications

On the rigorous derivation of the incompressible Euler equation from Newton’s second law

On the rigorous derivation of the incompressible Euler equation from Newton’s second law

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

A strong form of propagation of chaos for Cucker–Smale model

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