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Propagation of chaos for the 2D viscous vortex model

Mischler, Stéphane; Hauray, Maxime; Fournier, Nicolas (2014), Propagation of chaos for the 2D viscous vortex model, Journal of the European Mathematical Society, 16, 7, p. 1423–1466. http://dx.doi.org/10.4171/JEMS/465

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00762286
Date
2014
Journal name
Journal of the European Mathematical Society
Volume
16
Number
7
Publisher
European Mathematical Society
Pages
1423–1466
Publication identifier
http://dx.doi.org/10.4171/JEMS/465
Metadata
Show full item record
Author(s)
Mischler, Stéphane
Hauray, Maxime
Fournier, Nicolas
Abstract (EN)
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.
Subjects / Keywords
Entropy dissipation; Fisher information; Propagation of Chaos; Stochastic particle systems; 2D Navier-Stokes equation

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