Geometric Properties of the Maxwell Set and a Vortex Filament Structure for Burgers Equation

The inviscid limit of the stochastic Burgers equation 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. We examine the geometry of the Maxwell set in terms of the behaviour of the pre-Maxwell set, the pre-caustic and the pre-level surfaces. In particular, contrary to the ideas of Helmholtz and Lord Kelvin, we prove that even if initially the fluid flow is irrotational, in the inviscid limit, associated with the advent of the Maxwell set a non-zero vorticity vector forms in the fluid with vortex lines on the Maxwell set. This suggests that in quite general circumstances for small viscosity there is a vortex filament structure near the Maxwell set for both deterministic and stochastic Burgers equations.