不可压Euler方程的图灵普适性和Moore的一个猜想
Turing universality of the incompressible Euler equations and a conjecture of Moore
摘要：
本文构造了一个紧致的高维黎曼流形，在此流形上不定常Euler方程是Turing完备的.更准确地说，任何图灵机在给定输入下的停止等价于欧拉方程的某个全局解进入无发散向量场空间中的某个开集。特别地，这意味着具有初始数据的欧拉方程的解在无发散场的空间中是否会达到某个开集的不可判定性。这一结果进一步推动了陶的计划，即利用流体计算机研究欧拉和纳维尔-斯托克斯方程的爆破问题。作为一个显着的副产品，我们的证明方法使我们能够给一个反例的猜想摩尔追溯到1998年的不存在的解析映射的紧凑的流形是图灵完备的。
英文摘要：
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.
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