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Necessary and sufficient conditions for the third order Kolmogorov universalscaling flux laws are derived for the stochastically forced incompressibleNavier-Stokes equations on the torus in 2d and 3d. This paper rigorouslygeneralizes the result of \cite{bedrossian2019sufficient} to functions whichare heavy-tailed in Fourier space or have local finite time singularities inthe inviscid limit. In other words we have rigorously derived the well knownphysical relationship the direct cascade is a local process and is non-trivialif and only if energy moves toward the small scales or singularities haveoccurred. Similarly, an inverse cascade occurs if and only if energy movestowards the $k = 0$ Fourier mode in the invisicid limit.

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