STOCHASTIC PROCESSES ON GROUPS OF DIFFEOMORPHISMS AND VISCOUS HYDRODYNAMICS

Abstract

The viscous hydrodynamics is investigated via the studying diffusion processes on groups of Hs Sobolev diffeomorphisms of a flat n-dimensional torus (s > ½n + 1). A certain stochastic perturbation of the curve on the above groups, describing the motion of perfect incompressible fluid (or of the diffuse matter), is constructed such that it satisfies a certain stochastic analogue of the geodesic equation and the expectation of its "backward velocity" in the tangent space at identical diffeomorphism (algebra of the group) is a solution of Navier–Stokes (or Burgers, respectively) equations. In particular, the existence of solutions of those equations for lower s is proved. Some other approaches to stochastic presentation of viscous hydrodynamics, using the groups of diffeomorphisms, are also discussed.