Constructing Turing complete Euler flows in dimension $3$

TL;DR

This paper constructs a Turing complete stationary Euler flow on a 3D Riemannian manifold, advancing the understanding of computational universality in fluid dynamics and its implications for the Navier-Stokes blow-up problem.

Contribution

It provides the first known example of a Turing complete stationary Euler flow in three dimensions, addressing a longstanding open problem.

Findings

Constructed a Turing complete Euler flow on Riemannian S^3.

Shows implications for undecidability in fluid dynamics.

Connects to Tao's approach to Navier-Stokes blow-up problem.

Abstract

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.