# A kinetic selection principle for curl-free vector fields of unit norm

## TL;DR

This paper extends previous results on Lipschitz regularity of planar vector fields satisfying kinetic equations to those valued in spheres of general norms, establishing Hölder continuity and describing vortex singularities.

## Contribution

It generalizes the regularity results to arbitrary smooth, convex norms of power type, and characterizes the behavior of vector fields around vortex singularities.

## Key findings

- Vector fields are locally /(p-1)-Hf6lder continuous.
- Complete description of vector field behavior near vortex singularities.
- Rules out line-like singularities for curl-free vector fields in general norm spheres.

## Abstract

This article is devoted to the generalization of results obtained in 2002 by Jabin, Otto and Perthame. In their article they proved that planar vector fields taking value into the unit sphere of the euclidean norm and satisfying a given kinetic equation are locally Lipschitz. Here, we study the same question replacing the unit sphere of the euclidean norm by the unit sphere of \emph{any} norm. Under natural asumptions on the norm, namely smoothness and a qualitative convexity property, that is to be of power type $p$, we prove that planar vector fields taking value into the unit sphere of such a norm and satisfying a certain kinetic equation are locally $\frac{1}{p-1}$-H\"older continuous. Furthermore we completely describe the behaviour of such a vector field around singular points as a \emph{vortex} associated to the norm. As our kinetic equation implies for the vector field to be curl-free, this can be seen as a selection principle for curl-free vector fields valued in spheres of general norms which rules out line-like singularities.