Universality of Euler flows and flexibility of Reeb embeddings

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [38], [39] 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.
Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R& D (MDM-2014- 0445) via an FPI grant. Robert Cardona and Eva Miranda are supported by the grant PID2019-103849GBI00 of MCIN/AEI/10.13039/501100011033 and AGAUR grant 2021 SGR 00603. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via the ICREA Academia Prize 2016 and ICREA Academia Prize 2021 and by a Friedrich Wilhelm Bessel Research Award of the Alexander von Humboldt Foundation. She is also partially supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M). Daniel Peralta-Salas is supported by the grant PID2019-106715GB GB-C21 funded by MCIN/AEI/10.13039/501100011033. Robert Cardona, Eva Miranda and Daniel Peralta-Salas acknowledge partial support from the grant “Computational, dynamical and geometrical complexity in fluid dynamics”, Ayudas Fundación BBVA a Proyectos de Investigación Científica 2021. Francisco Presas is supported by the grant PID2019-108936GB-C21. Daniel Peralta-Salas and Francisco Presas also acknowledge partial support from the ICMAT-Severo Ochoa grant CEX2019-000904-S.
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Postprint (published version)