Stochastic particle approximations for two-dimensional Navier-Stokes equations

We present a probabilistic interpretation of some Navier-Stokes equations which describe the behaviour of the velocity field in a viscous incompressible fluid. We deduce from this approach stochastic particle approximations, which justify the vortex numerical schemes introduced by Chorin to simulate the solutions of the Navier-Stokes equations. After some recalls on the McKean-Vlasov model, we firstly study a Navier-Stokes equation defined on the whole plane. The probabilistic approach is based on the vortex equation, satisfied by the curl of the velocity field. The equation is then related to a nonlinear stochastic differential equation, and this allows us to construct stochastic interacting particle systems with a "propagation of chaos" property: the laws of their empirical measures converge, as the number of particles tends to infinity to a deterministic law with time-marginals solving the vortex equation. Our approach is inspired by Marchioro and Pulvirenti [26] and we improve their results in a pathwise sense. Next we study the case of a Navier-Stokes equation defined on a bounded domain, with a no-slip condition at the boundary. In this case, the vortex equation satisfies a Neumann condition at the boundary, which badly depends on the solution. We simplify the model by studying in details the case of a fixed Neumann condition and we finally explain how the results should be adapted in the Navier-Stokes case.