Uniform approximation of 2$d$ Navier-Stokes equation by stochastic interacting particle systems

TL;DR

This paper demonstrates that a system of stochastic differential equations modeling interacting particles can uniformly approximate the 2D Navier-Stokes equation in vorticity form, using a semigroup approach.

Contribution

It introduces a novel stochastic particle system that converges uniformly to the 2D Navier-Stokes solution, providing a new probabilistic approximation method.

Findings

Empirical process converges uniformly in time and space to Navier-Stokes solution.

Uses a mollified empirical process for convergence.

Employs a semigroup approach for proofs.

Abstract

We consider an interacting particle system modeled as a system of $N$ stochastic differential equations driven by Brownian motions. We prove that the (mollified) empirical process converges, uniformly in time and space variables, to the solution of the two-dimensional Navier-Stokes equation written in vorticity form. The proofs follow a semigroup approach.