We will briefly review Kolmogorov’s (41) theory of homogeneous turbulence and Onsager’s ( 49)
conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of
vanishing viscosity. Although over the past 60 years there is a vast body of literature related to this subject, at present
there is no rigorous mathematical proof that the solutions to the Navier-Stokes equations yield
Kolmogorov’s laws. For this reason various models have been introduced that are more tractable but capture
some of the essential features of the Navier-Stokes equations themselves. We will discuss one such
stochastically driven dyadic model for turbulent energy cascades. We will describe how results for stochastic PDEs
can be used to prove that this dyadic model is consistent with Kolmogorov’s theory and Onsager’s conjecture.

This is joint work with Vlad Vicol and Nathan Glatt-Holtz.