Generalized stochastic Lagrangian paths for the Navier-Stokes equation

TL;DR

This paper introduces a class of stochastic processes on Riemannian manifolds that characterize solutions to the Navier-Stokes equation using a generalized kinetic energy framework, extending classical and probabilistic approaches.

Contribution

It defines semimartingales on manifolds as critical points of kinetic energy, linking them to weak solutions of the Navier-Stokes equation with Ebin-Marsden's Laplacian.

Findings

Semimartingales are critical points iff their drifts solve Navier-Stokes.

Classical solutions minimize kinetic energy on the torus.

Extension of Brenier's generalized flows to Navier-Stokes.

Abstract

In the note added in proof of the seminal paper [Groups of diffeomorphisms andthe motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebinand Marsden introduced the so-called correct Laplacian for the Navier-Stokes equationon a compact Riemannian manifold. In the spirit of Brenier's generalized flows forthe Euler equation, we introduce a class of semimartingales on a compact Riemannianmanifold. We prove that these semimartingales are critical points to the correspondingkinetic energy if and only if its drift term solves weakly the Navier-Stokes equationdefined with Ebin-Marsden's Laplacian. We also show that for the torus case,classical solutions of the Navier-Stokes equation realize the minimum of the kineticenergy in a suitable class.