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

作者：Foias Ciprian^*; Rosa Ricardo M S; Temam Roger

来源：Comptes Rendus Mathematique, 2010, 348(5-6): 347-353.

DOI：10.1016/j.crma.2009.12.018

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.

• 出版日期2010-3