构造$3$维的图灵完备Euler流
Constructing Turing complete Euler flows in dimension $3$
摘要：
每个物理系统都能模拟任何一台图灵机吗？这是一个与某些物理现象的不可判定性密切相关的经典问题。关于流体流动，摩尔在[15]中问流体力学是否能够进行计算。最近，陶发起了一个计划的基础上图灵完整性的欧拉方程，以解决爆破问题的纳维-斯托克斯方程。在这个方向上，一些物理系统的不可判定性近年来已经被研究，从量子能隙问题[7]到量子场论[11]。据我们所知，自20世纪90年代初Moore的工作以来，三维流体流动中不可判定粒子路径的存在性一直是一个难以捉摸的开放问题。本文在黎曼$S^3$上构造了一个Turing完全定常Euler流，并推测了它对Tao处理Navier-斯托克斯方程爆破问题的意义.
英文摘要：
Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme 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 [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian $S^3$ and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations.
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