Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus

TL;DR

This paper links the Navier-Stokes equations on a torus with forward-backward stochastic differential equations, providing new probabilistic representations of solutions in the context of diffeomorphism groups.

Contribution

It introduces a novel connection between Navier-Stokes solutions and FBSDEs on the diffeomorphism group of a torus, offering a new probabilistic framework.

Findings

Established a correspondence between Navier-Stokes solutions and FBSDEs on diffeomorphism groups

Constructed diffusion process representations of Navier-Stokes solutions

Provided a probabilistic approach to analyze fluid dynamics on a torus

Abstract

We establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier-Stokes equations in terms of diffusion processes.