MPEJ Volume 12, No. 2, 33 pp.
Received: Nov 15, 2005. Revised: Mar 19, 2005. Accepted:  Apr 3, 2006.

S. Dostoglou, A. V. Fursikov, J. D. Kahl
Homogeneous and Isotropic Statistical Solutions of the Navier-Stokes equations

ABSTRACT: Two constructions of homogeneous and isotropic statistical solutions
of the 3D Navier-Stokes system are presented. First, homogeneous and isotropic
probability measures supported by weak solutions of the Navier-Stokes system
are produced by averaging over rotations the known homogeneous probability
measures, supported by such solutions, of \cite{VF1},  \cite{VF}.  It is then
shown how to approximate (in the sense of convergence of characteristic
functionals) any isotropic measure on a certain space of  vector fields by
isotropic measures supported by periodic vector fields and their rotations.
This is achieved without loss of uniqueness for the Galerkin system, allowing
for the Galerkin approximations of homogeneous statistical  Navier-Stokes
solutions to be adopted to isotropic approximations. The construction of
homogeneous measures in \cite{VF1}, \cite{VF} then applies to  produce
homogeneous and isotropic probability measures, supported by weak solutions of
the Navier-Stokes equations. In both constructions, the restriction of the
measures at $t=0$ is well defined and coincides with the initial measure.


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