Abstract

The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flow, there are no rigorous results confirming or infirming those predictions. Here, we consider a well-known simplified model for 3D turbulence: Burgulence, or turbulence for the 1D Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space force, we give sharp estimates for small-scale quantities such as increments and energy spectrum. Namely, we confirm results of the article of Aurell, Frisch, Lutsko and Vergassola published in 1991, using some of their ideas. We use SDE techniques (infinite-dimensional Itô calculus) as well as classical PDE techniques (maximum principle).