Generalized stochastic flows and applications to incompressible viscous fluids

Alexandra Antoniouk (1) , Marc Arnaudon (2) , Ana Bela Cruzeiro (3)

1 Dep. Nonlinear Analysis Institute of Mathematics NAS Ukraine
2 IMB - Institut de Mathématiques de Bordeaux
3 grupo de fisica matematica

We introduce a notion of generalized stochastic flows on manifolds, 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 $L^2$ 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 construct generalized flows with prescribed drift and kinetic energy smaller than the $L^2$ norm of the drift. The results are actually presented for general $L^q$ norms, thus including not only the Navier-Stokes equations but also other equations such as the porous media.

Domaines: Probabilités [math.PR]

Alexandra Antoniouk, Marc Arnaudon, Ana Bela Cruzeiro. Generalized stochastic flows and applications to incompressible viscous fluids. Bulletin des Sciences Mathématiques, 2014, 138 (4), pp.565-584. ⟨hal-01057528⟩