Looking at Euler flows through a contact mirror: Universality and Turing completeness

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, we show that the stationary Euler equations exhibit several universality features, In the sense 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. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Sullivan, Etnyre and Ghrist more than two decades ago.

We end up this talk addressing an apparently different question: What kind of physics might be non-computational? Using the former universality result, we can 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.. But, in view of the increase of dimension yielded by our proof. The question is can this be done in dimension 3? We will prove the existence of Turing complete fluid flows on a 3-dimensional geometric domain. Our novel strategy uses the computational power of symbolic dynamics and the contact mirror again.

This talk is based on joint work with Robert Cardona and Fran Presas ( arXiv:1911.01963 and arXiv:2012.12828 )

dynamical systems

Audience: researchers in the topic

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