Kolmogorov, Onsager and a stochastic model for turbulence

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.