On the stochastic Burgers equation with some applications to turbulence and astrophysics

TL;DR

This paper reviews results on the stochastic Burgers equation's inviscid limit, highlighting geometric structures and their relevance to turbulence and cosmological models, with new formulas for mass distribution in shocks.

Contribution

It introduces new explicit formulas for mass distribution within shocks and connects vortex structures to the early universe's evolution.

Findings

Vortex filament structures form near the Maxwell set at small viscosities.

Geometric properties of caustics and Hamilton-Jacobi surfaces are characterized.

New explicit formulas for mass distribution in shocks are provided.

Abstract

We summarise a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence. We show that for small viscosities there exists a vortex filament structure near to the Maxwell set. We discuss how this vorticity is directly related to the adhesion model for the evolution of the early universe and include new explicit formulas for the distribution of mass within the shock.