Kinetic formulation of vortex vector fields

Abstract

This article focuses on gradient vector fields of unit Euclidean norm in ${ℝ}^N$. The stream functions associated to such vector fields solve the eikonal equation and the prototype is given by the distance function to a closed set. We introduce a kinetic formulation that characterizes stream functions whose level sets are either spheres or hyperplanes in dimension $N\ge 3$. Our main result proves that the kinetic formulation is a selection principle for the vortex vector field whose stream function is the distance function to a point.