Looking at Euler flows through a contact mirror: Universality, Turing completeness and undecidability
The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [6, 7, 8] 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 [1].
A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Etnyre and Ghrist in [4] more than two decades ago.
We end this talk addressing a question raised by Moore in [5] : “Is hydrodynamics capable of performing computations?”. The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere. Can this result be improved? In [2] we construct a
Turing complete steady Euler flow in dimension 3. Time permitting, we discuss this and other generalizations for t-dependent Euler flows contained in [3].
In all the constructions above, the metric is seen as an additional "variable" and thus the method of proof does not work if the metric is prescribed.
Is it still possible to construct a Turing complete Euler flow on a 3-dimensional space with the standard metric? See the talk of Daniel Peralta-Salas in this workshop.
This talk is based on several joint works with Cardona, Peralta-Salas and Presas.
[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb embeddings, arXiv:1911.01963.
[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in dimension 3. PNAS May 11, 2021 118 (19) e2026818118; https://doi.org/10.1073/pnas.2026818118.
[3] R. Cardona, E. Miranda and D. Peralta-Salas, Turing universality of the incompressible Euler equations and a conjecture of Moore, International Mathematics Research Notices, rnab233, https://doi.org/10.1093/imrn/rnab233
[4] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture. Nonlinearity 13 (2000) 441–458.
[5] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991) 199–230.
[6] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.
[7] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete Cont. Dyn. Sys. A 38 (2018) 1553–1565.
[8] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.