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 and the other by Bertozzi-Constantin. In fact, Chemin proved a more general result, showing that vorticity initially having discontinuities only in directions normal to a family of vector fields continue to be so characterized by the time-evolved vector fields. A different four-page elementary proof of the regularity of a vortex patch boundary was published in 1994 by Serfati, employing only one vector field to describe the discontinuities in the initial data. In this talk, we discuss Serfati's proof along with a natural extension of it to a family of vector fields that reproduces the 1995 result of Chemin.