The K41 theory and turbulence in 1D Burgers equation

This work is a review of a group of results on the stochastic Burgers equation with small viscosity, obtained during the last two decades. These results jointly show that the equation makes a good model of hydrodynamical turbulence. The model provides natural and rigorously justified analogies of a number of key predictions of the theory of turbulence, including the main assertions of the Kolmogorov approach to turbulence, known as the K41 theory. The presentation is based on the recent book by Boritchev and Kuksin, One-Dimensional Turbulence and the Stochastic Burgers Equation (AMS, 2021) and on two papers that develop the book's results. Part of the assertions that we review here together with their rigorous proofs were earlier heuristically proved in works of physicists.