Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Probabilistic interpretation and particle method for vortex equations with Neumann's boundary condition


Code(s) de Classification MSC:

Résumé: We are interested in proving the convergence of Monte-Carlo approximations for vortex equations in bounded domains of $R^2$ with Neumann's condition on the boundary. This work is the first step to justify theorically some numerical algorithms for Navier-Stokes equations in bounded domains with no-slip conditions. We prove that the vortex equation has a unique solution in an appropriate space and can be interpreted in a probabilistic point of view through a nonlinear reflected process with space-time random births on the boundary of the domain. Next, we approximate the solution $w$ of this vortex equation by the weighted empirical measure of interacting diffusive particles with normal reflecting boundary conditions and space-time random births on the boundary. The weights are related to the initial data and to the Neumann condition. We can deduce from this result a simple stochastic particle algorithm to approximate $w$.

Mots Clés: Vortex equation on a bounded domain ; Monte-Carlo approximation ; Interacting particle systems with reflection ; space-time random births

Date: 2002-06-27

Prépublication numéro: PMA-742

Pdf file : PMA-742.pdf