Stochastic formulation of incompressible fluid flows in wall bounded regions

TL;DR

This paper develops a mathematical framework using stochastic differential equations to enable Monte-Carlo simulations of incompressible and turbulent wall-bounded fluid flows, facilitating probabilistic analysis of such complex systems.

Contribution

It introduces a novel probabilistic formulation of the Navier-Stokes equations using McKean-Vlasov SDEs, specifically tailored for wall-bounded turbulent flows.

Findings

Path integral representations for parabolic equations derived

Probabilistic formulations for Navier-Stokes equations established

Framework enables Monte-Carlo simulations of wall-bounded turbulence

Abstract

In this paper we establish a mathematical framework which may be used to design Monte-Carlo simulations for a class of time irreversible dynamic systems, such as incompressible fluid flows, including turbulent flows in wall-bounded regions, and some other (non-linear) dynamic systems. Path integral representations for solutions of forward parabolic equations are obtained, and, in combining with the vorticity transport equations, probabilistic formulations for solutions of the Navier-Stokes equations are therefore derived in terms of (forward) McKean-Vlasov stochastic differential equations (SDEs), which provides us with the mathematical framework for Monte-Carlo simulations of wall-bounded turbulent flows.