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A note on statistical solutions of the three-dimensional Navier-Stokes equations: The stationary case. (Note sur les solutions statistiques de équations de Navier-Stokes incompressibles en dimension trois d’espace: le cas stationnaire.) (English) Zbl 1186.35142
Summary: 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.

MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76D06 Statistical solutions of Navier-Stokes and related equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B41 Attractors 35B65 Smoothness and regularity of solutions to PDEs
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