A one dimensional analysis of singularities and turbulence for the stochastic Burgers equation in d-dimensions

TL;DR

This paper analyzes the inviscid limit of the stochastic Burgers equation in multiple dimensions, focusing on the geometric structures like caustics and Maxwell sets, and links their singularities to turbulent, stochastic behavior.

Contribution

It provides a one-dimensional reduction of the problem, explicit algebraic expressions for critical sets, and characterizes how their singularities induce stochastic turbulence.

Findings

Explicit algebraic surface for Maxwell set and caustic in polynomial case

Characterization of singular parts of caustics and Maxwell sets

Demonstration of rapid geometric changes causing stochastic turbulence

Abstract

The inviscid limit of the stochastic Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We give an explicit expression for an algebraic surface containing the Maxwell set and caustic in the polynomial case. Those parts of the caustic and Maxwell set which are singular are characterised. We demonstrate how the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature, and we determine its intermittence in terms of the recurrent behaviour of two processes.