Turing universality of the incompressible Euler equations and a conjecture of Moore

TL;DR

This paper constructs a high-dimensional compact manifold where the Euler equations are Turing complete, linking fluid dynamics to computability theory and providing a counterexample to Moore's conjecture.

Contribution

It demonstrates Turing completeness of Euler equations on a compact manifold, advancing Tao's program on fluid computability and blow-up problems.

Findings

Euler equations are Turing complete on a constructed manifold

Undecidability results for Euler solutions' behavior

Counterexample to Moore's conjecture on analytic Turing complete maps

Abstract

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.