The stochastic Burgers equation with vorticity: Semiclassical asymptotic series solutions with applications

Abstract

We consider a stochastic Burgers type equation which incorporates a vector potential. The solution of this equation is not of gradient form and so this equation can be described as a stochastic Burgers equation with vorticity. Building on previous work on the standard stochastic Burgers equation, we discuss the related Hamilton-Jacobi theory in detail and show how to find semiclassical series expansions for our stochastic Burgers equation with vorticity. We examine the behaviour of the solution in the inviscid limit and discuss the geometric structure of the resulting singularities. We illustrate these results with an example of a Burgers type fluid in a rotating bucket under a harmonic oscillator potential. We conclude with a discussion of the relationship between the correspondence limit of Nelson's stochastic mechanics and stationary state solutions for Burgers equations and illustrate an example of the equations studied in this paper arising from considerations of the Coulomb potential.