A random space-time birth particle method for 2d vortex equations with external field

Abstract

We consider incompressible 2d Navier-Stokes equations in the whole plane with external nonconservative forces fields. The initial data and external field are functions assumed to satisfy only slight integrability properties. We develop a probabilistic interpretation of these equations based on the associated vortex equation, in order to construct a numerical particle method to approximate the solutions. More precisely, we relate the vortex equation with additional term to a nonlinear process with random space-time birth, which provides a probabilistic description of the creation of vorticity. We then introduce interacting particle systems defined for a regularized interaction kernel, whose births are chosen randomly in time and space. By a coupling method, we show that these systems are approximations of the nonlinear process and obtain precise convergence estimates. From this result, we deduce a stochastic numerical particle method to obtain the vorticity and also to recover the velocity field. The results are either pathwise or of weak convergence, depending on the integrability of the data. We illustrate our results with simulations.