Turing universality of the incompressible Euler equations and a conjecture of Moore
In this article we construct a compact Riemannian manifold of high dimension on which the time dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open...
| Autores: | , , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/365673 |
| Acceso en línea: | https://hdl.handle.net/2117/365673 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Dynamical systems Analysis of PDEs Computational Complexity Dynamical Systems Turing universality Euler equations Conjecture of Moore Equacions en derivades parcials Sistemes dinàmics diferenciables Classificació AMS::35 Partial differential equations Classificació AMS::37 Dynamical systems and ergodic theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
| Sumario: | In this article we construct a compact Riemannian manifold of high dimension on which the time dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of wether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao’s programme to study the blowup problem for the Euler and Navier-Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete. |
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