A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case

Submitted by Jairo Amaro (jairo.amaro@sibi.ufrj.br) on 2019-07-09T17:19:34Z No. of bitstreams: 1 2010_ROSA_CRM_v348_p347-353-min.pdf: 154141 bytes, checksum: 75e7192cb3b34b1a821995550cabdaee (MD5)Made available in DSpace on 2019-07-09T17:19:34Z (GMT). No. of bitstreams: 1 2010_ROSA_CRM_v348_p347-353-min.pdf: 154141 bytes, checksum: 75e7192cb3b34b1a821995550cabdaee (MD5) Previous issue date: 2010-02-24Indisponível.Stationary statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble average for turbulent flows in statistical equilibrium in time. They are also a generalization of the notion of invariant measure to the case of the three-dimensional Navier–Stokes equations, for which a global uniqueness result is not known to exist and a semigroup may not be well-defined in the classical sense. The two classical definitions of stationary statistical solutions are considered and compared, one of them being a particular case of the other and possessing a number of useful properties. Furthermore, the so-called time-average stationary statistical solutions, obtained as generalized limits of time averages of weak solutions as the averaging time goes to infinity are shown to belong to this more restrictive class. A recurrent type result is also obtained for statistical solutions satisfying an accretion condition. Finally, the weak global attractor of the three-dimensional Navier–Stokes equations is considered, and in particular it is shown that there exists a topologically large subset of the weak global attractor which is of full measure with respect to that particular class of stationary statistical solutions and which has a certain regularity property