Kolmogorov's theory of turbulence, the Landau criticism and their rigorous 1d versions

Kolmogorov's theory of turbulence contains two celebrated heuristic laws, related to the second and third moments of increments of the fluid's velocity field $u(t,x+r)-u(t,x)$ for "small but not too small $r$". Kolmogorov claimed that the moments as functions of $r$ have the form (universal pre-factor) $x$ (certain exponent of $|r|$). Landau's criticism was that the pre-factor indeed may be universal for the third moment, but not for the second. In my talk, I will explain that the two heuristic laws allow rigorous versions, related to a fictitious one-dimensional fluid, described by the Burgers equation. There - indeed - the pre-factor in the law for the third moment is explicit and universal, but that for the second moment cannot be such. I will explain the difference between the two moments which leads to this effect.