DNA seminar: Helicity: invariance, uniqueness and dynamics

  • Date:
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64229
  • Lecturer: Daniel Peralta-Salas (Instituto de Ciencias Matemáticas, Spain)
  • Contact person: Michael Benedicks
  • Seminarium

Abstract: Helicity is a remarkable functional acting on the space of exact divergence-free vector fields on a Riemannian 3-manifold. It was introduced by Woltjer in the context of MHD and by Moreau and Moffatt in the context of fluid dynamics. Arnold showed that it defines an integral invariant under any kind of volume-preserving diffeomorphisms (orientation-preserving), and that it is related to the asymptotic (and averaged) linking number of the integral curves of the vector field. A surprising connection with dynamics unveiled by Taubes is that an exact divergence-free vector field of nonzero helicity cannot be uniquely ergodic. The goal of this talk is to review these fascinanting properties and to show that, under some mild technical assumptions, the helicity is unique, in the sense that any smooth enough functional that is invariant under arbitrary volume-preserving
diffeomorphisms is a function of the helicity. This is based on joint work with A. Enciso and F. Torres de Lizaur.