Turing complete Navier-Stokes steady states via cosymplectic geometry

TL;DR

This paper demonstrates that certain steady-state solutions of the Navier-Stokes equations on specific Riemannian 3-manifolds can perform universal computation, linking fluid dynamics, geometry, and computation.

Contribution

It introduces a novel connection between cosymplectic geometry and Navier-Stokes solutions, showing universality is possible under mild geometric conditions.

Findings

Existence of Turing complete steady states on certain manifolds.

Universality is compatible with viscosity under specific geometric conditions.

Extension of Beltrami-Reeb correspondence to cosymplectic geometry.

Abstract

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.