Cardona Robert, Miranda Eva, Peralta-Salas Daniel, Presas Francisco
Laboratory of Geometry and Dynamical Systems, Department of Mathematics and IMTech, Universitat Politècnica de Catalunya (UPC), Barcelona 08028, Spain.
BGSMath Barcelona Graduate School of Mathematics, Centre de Recerca Matemàtica (CRM) Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain.
Proc Natl Acad Sci U S A. 2021 May 11;118(19). doi: 10.1073/pnas.2026818118.
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, 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 [Formula: see text] and speculate on its implications concerning Tao's approach to the blow-up problem in the Navier-Stokes equations.
每个物理系统都能模拟任何图灵机吗？这是一个经典问题，与某些物理现象的不可判定性密切相关。关于流体流动，摩尔[C. 摩尔，《物理评论快报》4，199 (1991)]曾问流体动力学是否能够进行计算。最近，陶哲轩发起了一个基于欧拉方程的图灵完备性的项目，以解决纳维 - 斯托克斯方程中的爆破问题。在这个方向上，近年来已经研究了一些物理系统的不可判定性，从量子间隙问题到量子场论。据我们所知，自20世纪90年代初摩尔的研究以来，三维流体流动中不可判定粒子路径的存在一直是一个难以捉摸的开放问题。在本文中，我们在黎曼流形[公式：见原文]上构造了一个图灵完备的定常欧拉流，并推测其对陶哲轩解决纳维 - 斯托克斯方程爆破问题方法的影响。
