Some statistical properties of the Burgers equation with white-noise initial velocity

Abstract

We revisit the one-dimensional Burgers equation in the inviscid limit for white-noise initial velocity. We derive the probability distributions of velocity and Lagrangian increments, measured on intervals of any length $x$. This also gives the velocity structure functions. Next, for the case where the initial density is uniform, we obtain the distribution of the density, over any scale $x$, and we derive the density two-point correlation and power spectrum. Finally, we consider the Lagrangian displacement field and we derive the distribution of increments of the Lagrangian map. We check that this gives back the well-known mass function of shocks. For all distributions we describe the limiting scaling functions that are obtained in the large-scale and small-scale limits. We also discuss how these results generalize to other initial conditions, or to higher dimensions, and make the connection with a heuristic multifractal formalism. We note that the formation of point-like masses generically leads to a universal small-scale scaling for the density distribution, which is known as the ``stable-clustering ansatz'' in the cosmological context (where the Burgers dynamics is also known as the ``adhesion model'').

29 pages, published in J. Stat. Phys

References in corpus (6)

• Burgers Turbulence
• The fine-grained phase-space structure of Cold Dark Matter halos
• Using the Zeldovich dynamics to test expansion schemes
• Statistical properties of the Burgers equation with Brownian initial velocity
• The global picture of self-similar and not self-similar decay in Burgers Turbulence
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• Density fields and halo mass functions in the Geometrical Adhesion toy Model
• Shock statistics in higher-dimensional Burgers turbulence
• Exact Results for Propagators in the Geometrical Adhesion Model
• Statistics of shocks in a toy model with heavy tails
• An invariant in shock clustering and Burgers turbulence