A variational principle for the Navier-Stokes equation

Abstract

Motivated from Arnold's variational characterization of the Euler equation in terms of geodesic families of diffeomorphisms, a variational principle for the motion of incompressible viscous fluids is presented. A volume preserving diffusion process with drift velocity field subject to the Navier-Stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations.