Statistical Theory for the Stochastic Burgers Equation in the Inviscid Limit

TL;DR

This paper develops a statistical framework for the stochastic Burgers equation in the inviscid limit, deriving master equations for key probability densities without closure assumptions, and analyzing shock-related structures to understand velocity field statistics.

Contribution

It introduces a closure-free, dimension reduction approach for the stochastic Burgers equation, linking shock structures to statistical properties and deriving scaling laws.

Findings

Derived master equations for velocity and shock statistics

Established a -7/2 power tail for velocity gradient PDF

Obtained bounds on tail decay of velocity gradient

Abstract

A statistical theory is developed for the stochastic Burgers equation in the inviscid limit. Master equations for the probability density functions of velocity, velocity difference and velocity gradient are derived. No closure assumptions are made. Instead closure is achieved through a dimension reduction process, namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field, here the shocks. Master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks, namely the points of shock generation and collisions. The scaling laws of the structure functions are derived through the analysis of the master equations. Rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints. We also establish that the probability density function of the velocity gradient has a left power tail with exponent -7/2.