The vortex patches of Serfati

Abstract

In 1993, two proofs of the persistence of regularity of the boundary of a classical vortex patch for the 2D Euler equations were published, one by Chemin (announced in 1991) the other by Bertozzi and Constantin. Chemin, in fact, proved a more general result, extending it further in 1995 showing, roughly, that vorticity initially having discontinuities only in directions normal to a family of vector fields that together foliate the plane continue to be so characterized by the time-evolved vector fields. A different, four-page "elementary" proof of Chemin's 1993 result was published in 1994 by Ph. Serfati, who also gave a fuller characterization of the velocity gradient's regularity. We give a detailed version of Serfati's proof along with an extension of it to a family of vector fields that reproduces the 1995 result of Chemin.