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Looking at Euler flows through a contact mirror: universality and undecidability
dc.contributor.author | Miranda Galcerán, Eva |
dc.contributor.author | Peralta-Salas, Daniel |
dc.contributor.author | Cardona, Robert |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtiques |
dc.date.accessioned | 2022-04-11T10:04:04Z |
dc.date.available | 2022-04-11T10:04:04Z |
dc.date.issued | 2022-07-08 |
dc.identifier.citation | Miranda, E.; Peralta-Salas, D.; Cardona, R. Looking at Euler flows through a contact mirror: universality and undecidability. 2022. DOI 10.48550/arXiv.2107.09471. |
dc.identifier.other | https://arxiv.org/abs/2107.09471 |
dc.identifier.uri | http://hdl.handle.net/2117/365642 |
dc.description.abstract | The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao’s approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian 3-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic |
dc.description.sponsorship | Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Mar´ıa de Maeztu Programme for Units of Excellence in R& D (MDM-2014-0445) via an FPI grant. Robert Cardona and Eva Miranda are partially supported by the grants PID2019-103849GB-I00 / AEI / 10.13039/501100011033 and the AGAUR grant 2017SGR932. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Daniel Peralta-Salas is supported by the grants MTM PID2019-106715GBC21 (MICINN) and Europa Excelencia EUR2019-103821 (MCIU). This work was partially supported by the ICMAT–Severo Ochoa grant CEX2019-000904-S. |
dc.language.iso | eng |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics |
dc.subject.lcsh | Differential equations, Partial |
dc.subject.lcsh | Hamiltonian systems |
dc.subject.other | Euler equations |
dc.subject.other | Reeb flows |
dc.subject.other | Turing completeness |
dc.subject.other | Universality |
dc.title | Looking at Euler flows through a contact mirror: universality and undecidability |
dc.type | External research report |
dc.subject.lemac | Hamilton, Sistemes de |
dc.subject.lemac | Equacions en derivades parcials |
dc.contributor.group | Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions |
dc.identifier.doi | 10.48550/arXiv.2107.09471 |
dc.subject.ams | Classificació AMS::35 Partial differential equations::35Q Equations of mathematical physics and other areas of application |
dc.subject.ams | Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems |
dc.rights.access | Open Access |
local.identifier.drac | 32416008 |
dc.description.version | Preprint |
local.citation.author | Miranda, E.; Peralta-Salas, D.; Cardona, R. |
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