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Two-dimensional unit-length vector fields of vanishing divergence
Radu Ignat1

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.
1:  LM-Orsay - Laboratoire de Mathématiques d'Orsay
Sobolev spaces – regularity – approximation – vortices – kinetic formulation – entropy.