  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.
