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

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.