The Navier-Stokes-$\alpha$ equation via forward-backward stochastic differential systems

TL;DR

This paper employs forward-backward stochastic differential systems to analyze the Navier-Stokes-$\alpha$ equation, establishing existence and uniqueness results for both 2D and higher-dimensional cases with different boundary conditions.

Contribution

It introduces a stochastic approach to the Navier-Stokes-$\alpha$ equation, deriving a Feynman-Kac formula for the vorticity and proving solution existence and uniqueness in various dimensions.

Findings

Global existence and uniqueness for 2D case with periodic boundary conditions.

Local existence and uniqueness for d-dimensional ($d\geq 3$) case.

Derivation of Feynman-Kac formula for the vorticity equation.

Abstract

In this paper, we use forward-backward stochastic differential systems to study the solution of two and d dimensional ($d\geq 3$) Navier-Stokes-$\alpha$ equation. For the two dimensional Navier-Stokes-$\alpha$ equation with space periodic boundary conditions, we derive the Feynmann-Kac formula associated with the vorticity equation and prove the global existence and uniqueness of the solution. For the d dimensional ($d\geq 3$) case, we prove the local existence and uniqueness of the solution in Sobolev space.