Universality of Euler flows and flexibility of Reeb embeddings

TL;DR

This paper demonstrates the universality and complexity of steady Euler flows on manifolds, showing they can encode Turing machines and exhibit flexible embedding behaviors, linking contact topology with hydrodynamics.

Contribution

It proves that stationary Euler equations exhibit universality features, including Turing completeness and flexible Reeb embeddings, advancing the understanding of Euler flow complexity.

Findings

Steady Euler flows can encode universal Turing machines.

Reeb embeddings exhibit a new h-principle flexibility.

Euler solutions can be arbitrarily complex and undecidable.

Abstract

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows,i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature.