Necessary and Sufficient Conditions for Kolmogorov's Flux Laws on ${\mathbb{T}}^2$ and ${\mathbb{T}}^3$

Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier-Stokes equations on the torus in 2d and 3d. This paper rigorously generalizes the result of \cite{bedrossian2019sufficient} to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words we have rigorously derived the well known physical relationship the direct cascade is a local process and is non-trivial if and only if energy moves toward the small scales or singularities have occurred. Similarly, an inverse cascade occurs if and only if energy moves towards the $k=0$ Fourier mode in the invisicid limit.