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On the helicity in 3D-periodic Navier-Stokes equations. II: The statistical case. (English) Zbl 1184.35239

Summary: We study the asymptotic behavior of the statistical solutions to the Navier-Stokes equations using the normalization map [C. Foias and J. C. Saut, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 1–47 (1987; Zbl 0635.35075)]. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also investigated. We connect our mathematical analysis with the empirical theory of decaying turbulence. With appropriate mathematically defined ensemble averages, the Kolmogorov universal features are shown to be transient in time. We provide an estimate for the time interval in which those features may still be present.

MSC:

35Q30 Navier-Stokes equations
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
76F02 Fundamentals of turbulence
76M35 Stochastic analysis applied to problems in fluid mechanics

Citations:

Zbl 0635.35075
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References:

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