Circulation and Energy Theorem Preserving Stochastic Fluids

TL;DR

This paper introduces stochastic fluid equations that extend Kelvin's circulation theorem to noisy flows, preserving circulation or energy properties in a stochastic setting, advancing the understanding of fluid dynamics under randomness.

Contribution

It presents a new class of stochastic fluid equations that preserve circulation or energy theorems, generalizing classical deterministic results to stochastic flows.

Findings

Stochastic Euler-Poincaré equations extend Kelvin's theorem to noisy flows.

Solutions may preserve circulation or energy, but not both simultaneously.

The framework unifies deterministic and stochastic fluid dynamics with conservation properties.

Abstract

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler-Poincar\'{e} and stochastic Navier-Stokes-Poincar\'{e} equations respectively. The stochastic Euler-Poincar\'{e} equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.