At the end of 2023, ‘Universality of Euler flows and flexibility of Reeb embeddings’ was published in Advances in Mathematics, signed by Daniel Peralta-Salas (ICMAT-CSIC), Francisco Presas (ICMAT), Robert Cardona (Universitat Politècnica de Catalunya, UPC) and Eva Miranda (UPC).

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Terry Tao launched a programme to study the dynamical universality and the Turing completeness of the Euler and the Navier-Stokes equations. 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.

Reference: Robert Cardona, Eva Miranda, Daniel Peralta-Salas, Francisco Presas, Universality of Euler flows and flexibility of Reeb embeddings, Advances in Mathematics, Volume 428, 2023, 109142, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2023.109142.