Time-dependent Navier–Stokes Turing completeness

Determine whether there exist nonstationary solutions of the incompressible Navier–Stokes equations on smooth compact Riemannian 3‑manifolds whose dynamics are Turing complete, in the sense that a universal Turing machine can be simulated by the flow: for each initial configuration and desired output, there is a computable initial point and a computable target set such that the machine halts with that output if and only if the positive‑time trajectory through the initial point intersects the target set.

Background

The paper constructs stationary (time-independent) solutions to the incompressible Navier–Stokes equations on Hodge-admissible Riemannian 3-manifolds that are Turing complete by leveraging a correspondence between harmonic vector fields and weak cosymplectic structures. This establishes computational universality for viscous steady flows across all viscosities on a broad geometric class of manifolds.

In contrast, prior work has achieved Turing complete time-dependent flows for the inviscid Euler equations, but it remains unknown whether analogous time-dependent (nonstationary) solutions exist for the viscous Navier–Stokes equations. Clarifying this would extend undecidability and computational universality beyond steady states to genuinely time-dependent viscous fluid dynamics.