Statistical properties of shocks in Burgers turbulence

We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, {Mathematical expression}, with Gaussian whitenoise initial data. This system was originally proposed by Burgers^[1] as a crude model of hydrodynamic turbulence, and more recently by Zel'dovich et al..^[12] to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relation P(s)∞s^1/2, s≪1 where P(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1-P(s)≤exp{-Cs³} for s≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.