“Constructing Turing Complete Euler Flows in Dimension 3”, Robert Cardona, Eva Miranda, Daniel Peralta-Salas, Francisco Presas2020-12-23 (computability)⁠:

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, Moore1991 asked 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 to quantum field theories.

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 1990s.

In this article we construct a Turing complete stationary Euler flow on a Riemannian S³ and speculate on its implications concerning Tao’s approach to the blow up problem in the Navier-Stokes equations.