Kinetic formulation of vortex vector fields

TL;DR

This paper introduces a kinetic formulation for gradient vector fields of unit norm in Euclidean space, characterizing stream functions with spherical or hyperplanar level sets, and identifies the distance function to a point as a unique vortex field.

Contribution

It develops a new kinetic formulation as a selection principle for vortex vector fields with specific geometric stream functions in higher dimensions.

Findings

Kinetic formulation characterizes stream functions with spherical or hyperplanar level sets.

Proves the distance function to a point uniquely corresponds to a vortex vector field under this formulation.

Establishes a connection between the kinetic approach and geometric properties of vector fields.

Abstract

This article focuses on gradient vector fields of unit Euclidean norm in $\mathbb{R}^N$ . 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 $N \geq 3$. 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.