Stochastic Interacting Particle Systems and Nonlinear Partial Differential Equations from Fluid Mechanics

Abstract

We derive stochastic particle approximations for two nonlinear partial differential equations from fluid mechanics: the porous medium equation and the three-dimensional Navier-Stokes equation. We associate interacting particle systems with these equations and obtain, when the number of particles tends to infinity, laws of large numbers for the empirical measures.<br /> In the first chapter we study a system of interacting diffusions and show that the empirical measure of the particle system tends to the solution of the porous medium equation when the number of particles tends to infinity. Moreover we prove propagation of chaos for this system: if initially the positions of the particles are independent and identically distributed, then they remain so - at least approximately - in the course of time. <br /> In the second chapter we consider a sequence of nonlinear stochastic differential equations and show that the distributions of the solutions converge to the solution of the viscous porous medium equation.<br /> The third chapter deals with a stochastic particle approximation for the three-dimensional Navier-Stokes equation. This equation is of a completely different type than the porous medium equation, so that it seems difficult to treat it with the methods of the first chapter. Nevertheless this is possible: we do not consider the Navier-Stokes equation directly, but instead the equation satisfied by the vorticity, and use the fact that the velocity can be recovered from the vorticity.