# Propagation of chaos for the 2D viscous vortex model

• Volume: 016, Issue: 7, page 1423-1466
• ISSN: 1435-9855

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## Abstract

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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.

## How to cite

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Fournier, Nicolas, Hauray, Maxime, and Mischler, Stéphane. "Propagation of chaos for the 2D viscous vortex model." Journal of the European Mathematical Society 016.7 (2014): 1423-1466. <http://eudml.org/doc/277488>.

@article{Fournier2014,
abstract = {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.},
author = {Fournier, Nicolas, Hauray, Maxime, Mischler, Stéphane},
journal = {Journal of the European Mathematical Society},
keywords = {2D Navier-Stokes equation; stochastic particle systems; propagation of Chaos; Fisher information; entropy dissipation; vorticity equation; 2D Navier-Stokes equation; vorticity equation; stochastic particle systems; propagation of chaos; Fisher information; entropy dissipation},
language = {eng},
number = {7},
pages = {1423-1466},
publisher = {European Mathematical Society Publishing House},
title = {Propagation of chaos for the 2D viscous vortex model},
url = {http://eudml.org/doc/277488},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Fournier, Nicolas
AU - Hauray, Maxime
AU - Mischler, Stéphane
TI - Propagation of chaos for the 2D viscous vortex model
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 7
SP - 1423
EP - 1466
AB - 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.
LA - eng
KW - 2D Navier-Stokes equation; stochastic particle systems; propagation of Chaos; Fisher information; entropy dissipation; vorticity equation; 2D Navier-Stokes equation; vorticity equation; stochastic particle systems; propagation of chaos; Fisher information; entropy dissipation
UR - http://eudml.org/doc/277488
ER -

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