Constructing Turing complete Euler flows in dimension 3

Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier¿Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of three-dimensional fluid flows has remained an elusive open problem since Moore¿s works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao¿s approach to the blow-up problem in the Navier¿Stokes equations.
Robert Cardona was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Program for Units of Excellence in R&D (MDM-2014-0445) via an FPI grant. R.C. and E.M. are partially supported by Grants MTM2015-69135-P/FEDER, the Spanish Ministry of Science and Innovation PID2019-103849GB-I00/AEI/10.13039/501100011033, and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR932. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. D.P.-S. is supported by MICINN Grant MTM PID2019-106715GB-C21 and MCIU Grant Europa Excelencia EUR2019-103821. F.P. is supported by MICINN/FEDER Grants MTM2016-79400-P and PID2019-108936GB-C21. This work was partially supported by ICMAT–Severo Ochoa Grant CEX2019-000904-S.