A One-Dimensional Analysis of Singularities and Turbulence for the Stochastic Burgers Equation in d Dimensions

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.