Generalized Navier-Stokes flows and applications to incompressible viscous fluids

TL;DR

This paper introduces a generalized notion of stochastic flows on manifolds, extending classical fluid models to viscous fluids, and proves the existence of minimal energy flows with prescribed boundary conditions.

Contribution

It extends Brenier's perfect fluid flow framework to viscous fluids by defining generalized stochastic flows with energy minimization properties.

Findings

Existence of energy-minimizing generalized flows with prescribed initial and final states.

Construction of flows with prescribed drift and lower kinetic energy than classical norms.

Framework applies to Navier-Stokes and other equations like porous media.

Abstract

We introduce a notion of generalized stochastic flows on mani- folds, that extends to the viscous case the one defined by Brenier for perfect fluids. Their kinetic energy extends the classical kinetic energy to Brownian flows, defined as the L2 norm of their drift. We prove that there exists a generalized flow which realizes the infimum of the kinetic energy among all generalized flows with prescribed initial and final configuration. We also con- struct generalized flows with prescribed drift and kinetic energy smaller than the L2 norm of the drift. The results are actually presented for general Lq norms, thus including not only the Navier-Stokes equations but also other equations such as the porous media.