Hydrodynamics, probability and the geometry of the diffeomorphisms group

TL;DR

This paper extends Arnold's geometric interpretation of Euler flow to Navier-Stokes equations by characterizing solutions as stochastic geodesics on the diffeomorphisms group, linking hydrodynamics, probability, and geometry.

Contribution

It introduces a novel stochastic geometric framework for Navier-Stokes solutions, broadening the understanding of fluid dynamics through differential geometry and probability theory.

Findings

Navier-Stokes solutions as stochastic geodesics

Generalization of Arnold's Euler flow description

New connections between hydrodynamics and geometry

Abstract

We characterize the solution of Navier-Stokes equation as a stochastic geodesic on the diffeomorphisms group, thus generalizing Arnold's description of the Euler flow.