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 ope...
| Autores: | , , |
|---|---|
| Tipo de recurso: | artículo |
| 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:2117/365194 |
| Acceso en línea: | https://hdl.handle.net/2117/365194 https://dx.doi.org/10.1093/imrn/rnab233 |
| Access Level: | acceso abierto |
| Palabra clave: | Computer science Riemannian manifold Euler equations Turing complete Informàtica Classificació AMS::68 Computer science::68Q Theory of computing Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica::Algorísmica i teoria de la complexitat |
| 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 whether 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 blow-up 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. |
|---|