A trajectorial proof of the vortex method for the two-dimensional Navier-Stokes equation

We consider the Navier-Stokes equation in dimension 2 and more precisely th e vortex equation satisfied by the curl of the velocity field. We show the relation between this equation and a nonlinear stochastic differential equa tion. Next we use this probabilistic interpretation to construct approximat ing interacting particle systems which satisfy a propagation of chaos prope rty: the laws of the empirical measures tend, as the number of particles te nds to infinity, to a deterministic law for which marginals are solutions o f the vortex equation. This pathwise result justifies completely the vortex method introduced by Chorin to simulate the solutions of the vortex equati on. Our approach is inspired by Marchioro and Pulvirenti and we improve the ir results in a pathwise sense.