Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Monte-Carlo approximations of the solution of the 2d Navier-Stokes equation with finite measure initial data

Auteur(s):

Code(s) de Classification MSC:

• 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}
Résumé: We are interested in proving stochastic approximations for a 2d Navier-Stokes equation with an initial data $u_0$ belonging to the Lorentz space $L^{2,\infty}$ and such that curl $u_0$ is a finite measure. Giga, Miyakawa and Osada \cite{Giga:88} proved that a solution $u$ exists and that $u=K*{\rm curl}\ u$, where $K$ is the Biot-Savart kernel and $v={\rm curl}\ u$ is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution $v$ of this vortex equation by a stochastic interacting particle system and deduce a Monte-Carlo approximation for a solution of the Navier-Stokes equation. That gives in this case a pathwise proof of the vortex algorithm introduced by Chorin and consequently generalizes the works of Marchioro-Pulvirenti \cite{Marchioro:82} and M\'el\'eard \cite{Meleard:98} obtained in the case of an initial measure with bounded density.