Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence. (English) Zbl 1090.76017

Summary: Some rigorous results connected with the conventional statistical theory of turbulence in both the two and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular forcing term and for arbitrary stationary statistical solutions.


76D05 Navier-Stokes equations for incompressible viscous fluids
76F05 Isotropic turbulence; homogeneous turbulence
35Q30 Navier-Stokes equations
76D06 Statistical solutions of Navier-Stokes and related equations
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
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