Abstract
We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,
, with Gaussian whitenoise initial data. This system was originally proposed by Burgers1 as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet 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 relationP(s)∞s1/2,s≪1 whereP(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 fors≫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.
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