# Generalized stochastic flows and applications to incompressible viscous fluids

Abstract : 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.
Type de document :
Article dans une revue
Bulletin des Sciences Mathématiques, Elsevier, 2014, 138 (4), pp.565-584
Domaine :

https://hal.archives-ouvertes.fr/hal-01057528
Contributeur : Marc Arnaudon <>
Soumis le : mardi 9 septembre 2014 - 16:35:03
Dernière modification le : lundi 13 octobre 2014 - 15:43:25
Document(s) archivé(s) le : mercredi 10 décembre 2014 - 10:40:40

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• HAL Id : hal-01057528, version 1

### Citation

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

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