A Hamiltonian Mean-Field System for the Navier-Stokes Equation

TL;DR

This paper introduces a Hamiltonian particle system that models the Navier-Stokes equation, preserving key physical properties and connecting to existing approaches, offering new insights into fluid dynamics modeling.

Contribution

It develops a Hamiltonian mean-field particle system that derives the Navier-Stokes equation via a stochastic McKean-Vlasov framework, maintaining physical symmetries.

Findings

Kelvin Circulation Theorem holds along stochastic paths

Energy dissipation properties are discussed

Model connects to existing fluid dynamics approaches

Abstract

We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean-Vlasov equation yields the incompressible Navier Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.