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Related Concept Videos

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Constructing Turing complete Euler flows in dimension 3.

Robert Cardona1,2, Eva Miranda3,4,5, Daniel Peralta-Salas6

  • 1Laboratory of Geometry and Dynamical Systems, Department of Mathematics and IMTech, Universitat Politècnica de Catalunya (UPC), Barcelona 08028, Spain.

Proceedings of the National Academy of Sciences of the United States of America
|May 5, 2021
PubMed
Summary
This summary is machine-generated.

This study demonstrates Turing completeness in fluid dynamics by constructing a computable Euler flow. This finding advances research into the undecidability of physical systems and the Navier-Stokes blow-up problem.

Keywords:
Beltrami flowTuring completecontact geometrygeneralized shiftsincompressible Euler equations

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Area of Science:

  • Computational Physics
  • Fluid Dynamics
  • Theoretical Computer Science

Background:

  • The question of whether physical systems can simulate Turing machines is linked to physical undecidability.
  • Moore questioned hydrodynamics' computational capabilities in 1991.
  • Tao's program explores Turing completeness of Euler equations for Navier-Stokes blow-up.

Purpose of the Study:

  • To investigate the existence of undecidable particle paths in 3D fluid flows.
  • To construct a Turing complete stationary Euler flow.
  • To explore implications for the Navier-Stokes blow-up problem.

Main Methods:

  • Construction of a Turing complete stationary Euler flow on a Riemannian manifold.
  • Theoretical analysis of fluid dynamics and computability.

Main Results:

  • Successfully constructed a Turing complete stationary Euler flow.
  • Demonstrated that certain 3D fluid flows can perform universal computation.

Conclusions:

  • The study provides the first known example of undecidable particle paths in 3D fluid dynamics.
  • The findings may offer new perspectives on Tao's approach to the Navier-Stokes equations' blow-up problem.