Propagation of chaos for the 2D viscous vortex model

TL;DR

This paper proves that a stochastic particle system modeling 2D viscous vortices converges to the Navier-Stokes solution, demonstrating propagation of chaos and entropy preservation despite the singular Biot-Savart kernel.

Contribution

It establishes the propagation of chaos for the 2D viscous vortex model with entropy and Fisher information bounds, handling the singular kernel without viscosity restrictions.

Findings

Empirical measure converges to Navier-Stokes solution.

Propagation of chaos is proven for the vortex system.

Convergence is entropic and strong, with no entropy loss.

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.