Steady Euler flows and Beltrami fields in high dimensions
arXiv:2003.08112 · doi:10.1017/etds.2020.124
Abstract
Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The constructed solutions are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.
arXiv comment21 pages, 5 figures. Minor corrections, final version to appear at Ergodic Theory and Dynamical Systems
References in corpus (11)
- The Seiberg-Witten equations and the Weinstein conjecture
- The Weinstein conjecture for stable Hamiltonian structures
- Contact Topology and Hydrodynamics
- Universality of Euler flows and flexibility of Reeb embeddings
- On the Weinstein conjecture in higher dimensions
- Contact structures on 5-manifolds
- A characterization of 3D steady Euler flows using commuting zero-flux homologies
- Euler flows and singular geometric structures
- A note on the stationary Euler equations of hydrodynamics
- On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows
- Notes On Open Book Decompositions For Engel Structures