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- W1971350376 abstract "We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, $$frac{partial }{{partial t}}uleft( {x,t} right) + frac{partial }{{partial x}}left( {frac{1}{2}uleft( {x,t} right)^2 } right) = 0$$ , 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'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)∞s 1/2,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{−Cs 3} 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." @default.
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- W1971350376 date "1995-08-01" @default.
- W1971350376 modified "2023-10-08" @default.
- W1971350376 title "Statistical properties of shocks in Burgers turbulence" @default.
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- W1971350376 doi "https://doi.org/10.1007/bf02104509" @default.
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