Universality of Euler flows and flexibility of Reeb embeddings

Abstract

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows,i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature.

27 pages, 3 figures, minor changes, accepted for publication at Advances in Mathematics

References in corpus (5)

• Constructing Turing complete Euler flows in dimension $3$
• Computability and Beltrami fields in Euclidean space
• Turing universality of the incompressible Euler equations and a conjecture of Moore
• On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows
• Steady Euler flows and Beltrami fields in high dimensions

Cited by in corpus (11)

• Constructing Turing complete Euler flows in dimension $3$
• Computability and Beltrami fields in Euclidean space
• Turing universality of the incompressible Euler equations and a conjecture of Moore
• Steady Euler flows and Beltrami fields in high dimensions
• Contact structures and Beltrami fields on the torus and the sphere
• On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
• An $h$-principle for embeddings transverse to a contact structure
• Towards a Fluid computer
• An equivariant Reeb-Beltrami correspondence and the Kepler-Euler flow
• The singular Weinstein conjecture
• A note on the Turing universality of homogeneous potential wells and geodesible flows