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Rigorous remarks about scaling laws in turbulent fluids. (English) Zbl 1140.76011
Summary: A definition of scaling law for suitable families of measures is given and investigated. First, a number of necessary conditions are proved. They imply the absence of scaling laws for 2D stochastic Navier-Stokes equations and for stochastic Stokes (linear) problem in any dimension, while they imply a lower bound on the mean vortex stretching in 3D. Second, for 3D stochastic Navier-Stokes equations, necessary and sufficient conditions for scaling laws to hold are given, translating the problem into bounds for energy and enstrophy of high and low modes respectively. Unlike in the 2D case, the validity or invalidity of such conditions in 3D remains open.

MSC:
76F20 Dynamical systems approach to turbulence
76D06 Statistical solutions of Navier-Stokes and related equations
35Q30 Navier-Stokes equations
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