Looking at Euler flows through a contact mirror: Universality and undecidability

Abstract

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers by Cardona, Miranda, and Peralta- Salas, several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao (2019) as a novel way to address the problem of global existence for Euler and Navier–Stokes. Universality of the Euler equations was proved by Cardona et al. (2019) for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao’s approach relating Turing machines to Navier– Stokes equations, a Turing complete stationary Euler solution on a Riemannian 3-dimensional sphere was constructed by Cardona et al. (2021). Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable (as shown by Turing (1936)), a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths (as shown by Cardona et al. (2021)). In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of the work of Cardona et al. (2021) allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic.