PROBABILISTIC INTERPRETATION AND PARTICLE METHOD FOR VORTEX EQUATIONS WITH NEUMANN’S BOUNDARY CONDITION

• Published in: Proceedings of the Edinburgh Mathematical Society Vol. 47; no. 3; pp. 597 - 624
• Main Authors: Jourdain, Benjamin, Méléard, Sylvie
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.2004
• Subjects: interacting particle systems with reflection; Monte Carlo approximation; space-time random births; vortex equation on a bounded domain; vortex equation on a bounded domain; Monte Carlo approximation; interacting particle systems with reflection; space-time random births
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091503000142

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