Lagrangian structures for the Stokes, Navier-Stokes and Euler equations

TL;DR

This paper demonstrates that the Navier-Stokes, Euler, and Stokes equations can be formulated as stochastic Lagrangian systems by embedding them into a stochastic variational framework, revealing their underlying Lagrangian structure.

Contribution

It introduces a stochastic embedding approach to establish a Lagrangian formulation for fundamental fluid dynamics equations, extending classical variational principles to stochastic systems.

Findings

Navier-Stokes, Euler, and Stokes equations admit a stochastic Lagrangian structure

These equations are extremals of an explicit stochastic Lagrangian functional

The approach extends classical variational principles to stochastic fluid dynamics

Abstract

We prove that the Navier-Stokes, the Euler and the Stokes equations admit a Lagrangian structure using the stochastic embedding of Lagrangian systems. These equations coincide with extremals of an explicit stochastic Lagrangian functional, i.e. they are stochastic Lagrangian systems in the sense of [Cresson-Darses, J. Math. Phys. 48, 072703 (2007]