1D and multi-d Burgers Turbulence as a model case for the Kolmogorov Theory

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 flows, 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 or multi-dimensional potential Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum.