On the 2/3- and 4/5-laws of the Kolmogorov theory of turbulence and their rigorous 1d versions

In three papers, published in 1941, A.N.Kolmogorov built his celebrated heuristic theory of turbulence. Postulating that the velocity u(t,x) of a turbulent flow is a random field, stationary in time and homogeneous in space, he examined the second and third moments of the increments u(t,x+r)-u(t,x), when the Reynolds number of the flow is large and the increments r are "short, but not too short". I will talk about fictitious 1d fluid, described by the stochastic Burgers equation, consider increments of its velocity field, and will rigorously derive for their second and third moments natural analogies of the corresponding laws from Kolmogorov's theory. The talk is based on results from my joint book with A.Boritchev (AMS, 2021) and on some their recent development.