Book
Looking at Euler flows through a contact mirror: Universality and undecidability
| Title: | Looking at Euler flows through a contact mirror: Universality and undecidability |
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| Authors: | Cardona Aguilar, Robert, Miranda Galcerán, Eva, Peralta-Salas, Daniel |
| Contributors: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions |
| Source: | European Congress of Mathematics ISBN: 9783985470518 UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Publication Status: | Preprint |
| Publisher Information: | European Mathematical Society - EMS - Publishing House GmbH, 2023. |
| Publication Year: | 2023 |
| Subject Terms: | Differential equations, FOS: Computer and information sciences, Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials [Àrees temàtiques de la UPC], Dynamical Systems (math.DS), Computational Complexity (cs.CC), Universality, 01 natural sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, Hamilton, and nonholonomic systems, Hamiltonian systems, Mathematics - Dynamical Systems, 0101 mathematics, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Lagrangian, 35 Partial differential equations::35Q Equations of mathematical physics and other areas of application [Classificació AMS], Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics, Equacions en derivades parcials, Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics [Àrees temàtiques de la UPC], 37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], Differential equations, Partial, Euler equations, Sistemes de, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials, Computer Science - Computational Complexity, Hamilton, Sistemes de, Mathematics - Symplectic Geometry, Reeb flows, Symplectic Geometry (math.SG), Turing completeness, Classificació AMS::35 Partial differential equations::35Q Equations of mathematical physics and other areas of application, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, contact, Partial, Analysis of PDEs (math.AP) |
| Description: | 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. 24 pages, 1 figure |
| Document Type: | Part of book or chapter of book Article Conference object Report |
| File Description: | application/pdf |
| Language: | English |
| DOI: | 10.4171/8ecm/31 |
| DOI: | 10.48550/arxiv.2107.09471 |
| Access URL: | http://arxiv.org/abs/2107.09471 https://arxiv.org/abs/2107.09471 https://hdl.handle.net/2117/393208 https://doi.org/10.4171/8ecm/31 https://hdl.handle.net/2117/365642 https://arxiv.org/abs/2107.09471 https://doi.org/10.48550/arxiv.2107.09471 |
| Rights: | arXiv Non-Exclusive Distribution CC BY NC ND |
| Accession Number: | edsair.doi.dedup.....b6812e7b667c1b5fdd2eec562f511e46 |
| Database: | OpenAIRE |
| DOI: | 10.4171/8ecm/31 |
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