A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case |
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Authors: | Ciprian Foias Ricardo MS Rosa Roger Temam |
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Institution: | 1. Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530 Ilha do Fundão, Rio de Janeiro, RJ 21945-970, Brazil;3. Académie des Sciences, Paris, et Department of Mathematics, Indiana University, Bloomington, IN 47405, USA |
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Abstract: | 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. |
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