A characterization of 3D steady Euler flows using commuting zero-flux homologies

Abstract

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

16 pages, we added proofs of analogous results in higher dimensions, and a characterization of 3-dimensional Reeb fields

References in corpus (1)

• Contact Topology and Hydrodynamics

Cited by in corpus (6)

• Constructing Turing complete Euler flows in dimension $3$
• Universality of Euler flows and flexibility of Reeb embeddings
• Existence of global symmetries of divergence-free fields with first integrals
• Steady Euler flows and Beltrami fields in high dimensions
• The periodic orbit conjecture for steady Euler flows
• A note on the Turing universality of homogeneous potential wells and geodesible flows