Abstract:
A volume-preserving vector field $X$ on a manifold $M$ is Eulerisable if there exists a Riemannian metric $g$ on $M$ such that $X$ satisfies the stationary Euler equations on $(M,g)$. In this talk I will review some recent results on the dynamics of Eulerisable flows. In the first part I will present a homological characterization which generalizes the classical one by Sullivan for geodesible flows; as an application, I will show that this result implies that the Eulerisable flows cannot exhibit (volume-preserving) plugs. This is based on joint work with Ana Rechtman and Francisco Torres de Lizaur. In the second part, I will show that the Eulerisable flows are universal in the sense of Tao, i.e., any non-autonomous dynamics is extendable to an Euler flow on a sphere of sufficiently high dimension for some Riemannian metric. This implies, in particular, the Turing completeness of the Euler fields.
This is based on a joint work with Robert Cardona, Eva Miranda and Francisco Presas.