Geometric properties of the Maxwell set and a vortex filament structure for Burgers equation
The inviscid limit (\(\mu\to 0\)) of the viscous stochastic Burgers equation \[ \frac{\partial v^\mu}{\partial t}+\left(v^\mu\cdot \nabla\right)v^\mu =\frac{\mu^2}{2}\Delta v^\mu -\nabla V(x)-\epsilon\nabla k_t(x)\dot W_t \] is discussed in terms of the level surfaces of the minimizing Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The geometry of the Maxwell set is examined in terms of the behaviour of the pre-Maxwell set, the pre-caustic and the pre-level surfaces. In particular it is proved 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.
scientific article; zbMATH DE number 5159810
scientific article