Remarks on the K41 scaling law in turbulent fluids

A definition of K41 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 the 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 K41 are proved, translating the problem into bounds for energy and enstrophy of high and low modes respectively. The validity of such conditions in 3D remains open. Finally, a stochastic vortex model with such properties is presented.