Homogeneous and Isotropic Statistical Solutions of the Navier-Stokes equations
- Autores: A. V. Fursikov, J.D. Kahl, Stamatis Dostoglou
- Localización: Mathematical Physics Electronic Journal, ISSN-e 1086-6655, Vol. 12, Nº 2, 2006
- Idioma: inglés
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Resumen
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.
