PROBABILISTIC INTERPRETATION AND PARTICLE METHOD FOR VORTEX EQUATIONS WITH NEUMANN’S BOUNDARY CONDITION - Jourdain, Benjamin; Méléard, Sylvie (Proceedings of the Edinburgh Mathematical Society, 2004)
We are interested in proving the convergence of Monte Carlo approximations for vortex equations in bounded domains of $\mathbb{R}^2$ with Neumann’s condition on the boundary. This work is the first step towards justifying theoretically 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 energy space and can be interpreted from 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 prove a trajectorial propagation-of-chaos result for these systems of interacting particles. We can deduce a simple stochastic particle algorithm to simulate $w$. AMS 2000 Mathematics subject classification: Primary 60K35; 76D05