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, Nonlinearity 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 S3 and speculate on its implications concerning Tao’s approach to the blow-up problem in the Navier–Stokes equations.

每个物理系统都可以模拟任何图灵机吗？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，Moore [C. 摩尔，非线性4, 199 (1991)] 询问流体动力学是否能够进行计算。最近，Tao 推出了一个基于欧拉方程的图灵完备性的程序，以解决 Navier-Stokes 方程中的爆炸问题。在这个方向上，近年来从量子间隙问题到量子场论，研究了一些物理系统的不可判定性。据我们所知，自 1990 年代初摩尔的工作以来，三维流体流动的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，我们在黎曼分布上构建了图灵完备的平稳欧拉流秒3 并推测其对道在纳维-斯托克斯方程中解决爆炸问题的方法的影响。