# Statistical properties of shocks in Burgers turbulence

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## Abstract

We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,\(\frac{\partial }{{\partial t}}u\left( {x,t} \right) + \frac{\partial }{{\partial x}}\left( {\frac{1}{2}u\left( {x,t} \right)^2 } \right) = 0\), 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{−*Cs*^{3}} 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.

## Keywords

Initial Data Brownian Motion Spatial Location Quantum Computing Cumulative Probability## Preview

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