Turing complete Navier-Stokes steady states via cosymplectic geometry

Søren Dyhr; Ángel González-Prieto; Eva Miranda; Daniel Peralta-Salas

doi:10.1093/pnasnexus/pgag131

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arXiv · 2507.07696

Turing complete Navier-Stokes steady states via cosymplectic geometry

Søren Dyhr·Ángel González-Prieto·Eva Miranda·Daniel Peralta-Salas

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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.

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Søren Dyhr, Ángel González-Prieto, Eva Miranda, Daniel Peralta-Salas. 2025-07-10. Turing complete Navier-Stokes steady states via cosymplectic geometry. https://doi.org/10.1093/pnasnexus/pgag131

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