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, 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently,...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 2021-05, Vol.118 (19), p.1
Main Authors: Cardona, Robert, Miranda, Eva, Peralta-Salas, Daniel, Presas, Francisco
Format: Article
Language:English
Subjects:
Computational fluid dynamics
Dynamical Systems
Euler-Lagrange equation
Fluid flow
Hydrodynamics
Mathematical analysis
Mathematics
Navier-Stokes equations
Nonlinear systems
Physical Sciences
Three dimensional flow
Turing machines
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Summary: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, 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 [Formula: see text] and speculate on its implications concerning Tao's approach to the blow-up problem in the Navier-Stokes equations.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.2026818118