On energy cascades in non-homogeneous 3D Navier-Stokes equations
  We show - in the framework of physical scales and $(K_1,K_2)$-averages - that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws - in terms of Grashof number - for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales. 
R. Dascaliuc and Z. Gruji\'c
math.AP math-ph math.MP physics.flu-dyn