1. Constructing Turing complete Euler flows in dimension 3
- Author
-
Francisco Presas, Robert Cardona, Eva Miranda, Daniel Peralta-Salas, Universitat Politècnica de Catalunya [Barcelona] (UPC), Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Lille-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Autonoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Carlos III de Madrid [Madrid] (UC3M), Ministerio de Economía y Competitividad (España), Ministerio de Ciencia e Innovación (España), Ministerio de Ciencia, Innovación y Universidades (España), Observatoire de Paris, Université Paris sciences et lettres (PSL), Universidad Autónoma de Madrid (UAM), Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions, Universidad Autonoma de Madrid (UAM), and Universidad Carlos III de Madrid [Madrid] (UC3M)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)
- Subjects
FOS: Computer and information sciences ,Generalized shifts ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,MathematicsofComputing_NUMERICALANALYSIS ,Mathematics::Analysis of PDEs ,Dynamical Systems (math.DS) ,Computational Complexity (cs.CC) ,01 natural sciences ,53 Differential geometry [Classificació AMS] ,Physics::Fluid Dynamics ,contact geometry ,Mathematics - Analysis of PDEs ,Political science ,Incompressible Euler equations ,0103 physical sciences ,FOS: Mathematics ,Incompressible euler equations ,Turing complete ,Mathematics - Dynamical Systems ,0101 mathematics ,[MATH]Mathematics [math] ,010306 general physics ,generalized shifts ,Multidisciplinary ,010102 general mathematics ,Matemàtiques i estadística [Àrees temàtiques de la UPC] ,language.human_language ,incompressible Euler equations ,Computer Science - Computational Complexity ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Mathematics - Symplectic Geometry ,Contact geometry ,Physical Sciences ,language ,Symplectic Geometry (math.SG) ,Catalan ,Christian ministry ,Humanities ,Beltrami flow ,Analysis of PDEs (math.AP) - 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, 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.
- Published
- 2021
- Full Text
- View/download PDF