Constructing Turing complete Euler flows in dimension 3.
basic_science · Level V
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- Record sourced from PubMed, PMID 33947820.
- Also identified by DOI 10.1073/pnas.2026818118 and PMC identifier 8126859.
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Abstract
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, <i>Nonlinearity</i> 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.