Turing complete Navier-Stokes steady states via cosymplectic geometry

Area of Science:

• Mathematical physics
• Differential geometry
• Computational theory

Background:

• The Navier-Stokes equations describe fluid motion and are fundamental in physics.
• Turing completeness signifies the capability of a system to perform any computation.
• Investigating computational universality in geometric settings is an emerging area.

Purpose of the Study:

• To construct stationary solutions to the Navier-Stokes equations on specific Riemannian three-manifolds.
• To demonstrate that these manifolds can exhibit Turing completeness.
• To explore the relationship between geometry, viscosity, and computational universality.

Main Methods:

• Utilizing a correspondence between nonvanishing harmonic one-forms and cosymplectic geometry.
• Extending the classical correspondence between Beltrami fields and Reeb flows.
• Constructing specific solutions on Riemannian manifolds satisfying cohomological conditions.

Main Results:

• Stationary solutions to the Navier-Stokes equations were successfully constructed.
• The studied Riemannian manifolds were shown to be Turing complete.
• Computational universality was demonstrated to be compatible with viscosity under certain geometric conditions.

Conclusions:

• Viscosity does not obstruct computational universality on Riemannian manifolds with nonvanishing harmonic one-forms.
• A mild cohomological condition on the manifold's geometry is sufficient for computational universality.
• The findings link fluid dynamics, differential geometry, and theoretical computer science.