A stochastic-Lagrangian approach to the Navier--Stokes equations in domains with boundary

TL;DR

This paper develops a probabilistic, stochastic-Lagrangian framework to represent the 3D Navier--Stokes equations in bounded domains, highlighting boundary effects on fluid velocity.

Contribution

It extends previous boundary-free formulations by incorporating boundary vorticity influence into a probabilistic representation of Navier--Stokes equations.

Findings

Probabilistic representation of Navier--Stokes with boundaries derived

Boundary vorticity impacts interior fluid velocity nonlocally

Framework generalizes boundary-free stochastic models

Abstract

In this paper we derive a probabilistic representation of the deterministic 3-dimensional Navier--Stokes equations in the presence of spatial boundaries. The formulation in the absence of spatial boundaries was done by the authors in [Comm. Pure Appl. Math. 61 (2008) 330--345]. While the formulation in the presence of boundaries is similar in spirit, the proof is somewhat different. One aspect highlighted by the formulation in the presence of boundaries is the nonlocal, implicit influence of the boundary vorticity on the interior fluid velocity.