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Turing universality of the incompressible Euler equations and a conjecture of Moore
arXiv - CS - Computational Complexity Pub Date : 2021-04-09 , DOI: arxiv-2104.04356
Robert Cardona, Eva Miranda, Daniel Peralta-Salas

In this article we construct a compact Riemannian manifold of high dimension on which the time dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao's programme to study the blow-up problem for the Euler and Navier-Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete.

中文翻译:

不可压缩的Euler方程的Turing普遍性和Moore的猜想

在本文中,我们构造了一个紧凑的高维黎曼流形,其上与时间有关的欧拉方程是图灵完备的。更准确地说,任何具有给定输入的图灵机的停止都等同于在无散度矢量场的空间中进入某个开放集合的Euler方程的某个整体解。特别是,这意味着无法确定具有初始基准的Euler方程的解在无散度场的空间中是否会达到某个开放集。这个结果在Tao的程序中更进一步,该程序使用流体计算机研究Euler和Navier-Stokes方程的爆炸问题。作为一项出色的分拆,
更新日期:2021-04-12
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