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

• Autor(a) principal: Foias, Ciprian
• Data de Publicação: 2010
• Outros Autores: Rosa, Ricardo Martins da Silva, Temam, Roger
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFRJ
• Texto Completo: http://hdl.handle.net/11422/8736
• topic: Navier–Stokes equations; Vishik–Fursikov stationary statistical; CNPQ::CIENCIAS EXATAS E DA TERRA::FISICA::AREAS CLASSICAS DE FENOMENOLOGIA E SUAS APLICACOES::DINAMICA DOS FLUIDOS

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.