On Self-Similar Evolution for Multi-Dimensional Burgers Turbulence

This work is devoted to the evolution of random solutions of the unforced Burgers equation in d dimensions (v is the velocity and ψ the potential) in the limit of vanishing viscosity ν. The one-dimensional “nonlinear diffusion equation” was originally introduced in the thirties by Jan M. Burgers as a model for turbulence. The three-dimensional Burgers equation has also received attention as an approximate model for the formation of large-scale structure of the Universe when pressure is negligible; it describes then the statistical properties of gravitational turbulence, that is, the nonlinear stage of the gravitational instability developing from random initial perturbations [1]–[3]. Other problems leading to multi-dimensional Burgers equations or variants include surface growth under deposition of dust and flame front motion. In such instances, the potential corresponds to the shape of the surface or of the front.