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

TL;DR

This paper analyzes the statistical properties of the one-dimensional Burgers equation with white-noise initial velocity, deriving distributions and correlations for velocity, density, and Lagrangian fields, and exploring their scaling behaviors.

Contribution

It provides explicit probability distributions and scaling functions for various fields in the Burgers equation with white-noise initial conditions, extending understanding of shock formation and multifractal properties.

Findings

Derived distributions for velocity and density increments.

Established scaling functions in large-scale and small-scale limits.

Connected results to the stable-clustering hypothesis in cosmology.

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'').