A stochastic-Lagrangian particle system for the Navier-Stokes equations

TL;DR

This paper introduces a stochastic-Lagrangian particle system for Navier-Stokes equations, proves its global existence in 2D, and demonstrates convergence to the classical solution as the number of particles increases.

Contribution

It develops a particle-based stochastic formulation of Navier-Stokes, establishes global solutions in 2D, and shows convergence of the system to the classical equations as particle number grows.

Findings

Global solutions exist in 2D for initial data in Hölder spaces.

The system converges to Navier-Stokes solutions as N approaches infinity.

For fixed N, the energy decays roughly by 1/N over time.

Abstract

This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space $\holderspace{1}{\alpha}$ which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly.