Generalized stochastic Lagrangian paths for the Navier-Stokes equation

Marc Arnaudon (1) , Ana Bela Cruzeiro (2) , Shizan Fang (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 grupo de fisica matematica
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]

The purpose of this work is to establish a counterpart for the Navier-Stokes equation of Brenier's generalized flows for the Euler equation. We introduce a class of semi-martingales on a compact Riemannian manifold which will be generalized solutions of the Navier-Stokes equation. We prove that these semimartingales are critical points to the corresponding kinetic energy and that classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.

Domaines: Probabilités [math.PR]

Marc Arnaudon, Ana Bela Cruzeiro, Shizan Fang. Generalized stochastic Lagrangian paths for the Navier-Stokes equation. 2015. ⟨hal-01197123v1⟩