Propagation of chaos for the 2D viscous vortex model - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of the European Mathematical Society Année : 2014

Propagation of chaos for the 2D viscous vortex model

Résumé

We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly stronger result : the propagation of chaos of the stochastic paths towards the solution of the expected nonlinear stochastic differential equation. Moreover, the convergence holds in a strong sense, usually called entropic (there is no loss of entropy in the limit). The result holds without restriction (but positivity) on the viscosity parameter. The main difficulty is the presence of the singular Biot-Savart kernel in the equation. To overcome this problem, we use the dissipation of entropy which provides some (uniform in $N$) bound on the Fisher information of the particle system, and then use extensively that bound together with classical and new properties of the Fisher information.
Fichier principal
Vignette du fichier
ChaosVortex.pdf (484.49 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00762286 , version 1 (06-12-2012)

Identifiants

Citer

Nicolas Fournier, Maxime Hauray, Stéphane Mischler. Propagation of chaos for the 2D viscous vortex model. Journal of the European Mathematical Society, 2014. ⟨hal-00762286⟩
248 Consultations
398 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More