论文标题
建造图灵完整的欧拉流动$ 3 $
Constructing Turing complete Euler flows in dimension $3$
论文作者
论文摘要
每个物理系统都可以模拟任何图灵机吗？这是一个经典的问题，与某些物理现象的不可证明性密切相关。关于流体流，摩尔在[15]中询问流体动力学是否能够执行计算。最近，Tao根据Euler方程的Turing完整性启动了一个程序，以解决Navier-Stokes方程中的爆炸问题。在这个方向上，近年来已经研究了某些物理系统的不可证明性，从量子间隙问题[7]到量子场理论[11]。据我们所知，自1990年代初期摩尔的作品以来，3D流体流的不可发现的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Riemannian $ s^3 $上构建了Turing完整的固定Euler流，并推测了其关于Tao在Navier-Stokes方程中对爆炸问题的方法的含义。
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.
