Two-dimensional unit-length vector fields of vanishing divergence

Radu Ignat (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We prove the following regularity result: any two-dimensional unit-length divergence-free vector field belonging to $H^{1/2}$ (or $W^{1,1}$) is locally Lipschitz except at a locally finite number of vortices. We also prove approximation results for such vector fields: the dense sets are formed either by unit-length divergence-free vector fields that are smooth except at a finite number of points and the approximation result holds in the $W_{loc}^{1,p}$-topology ($1\leq p<2$), or by everywhere smooth unit-length vector fields (not necessarily divergence-free) and the approximation result holds in a weaker topology.

Mots clés: Sobolev spaces; regularity; approximation; vortices; kinetic formulation; entropy; entropy.

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Radu Ignat. Two-dimensional unit-length vector fields of vanishing divergence. 2011. ⟨hal-00601054⟩