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SUMMARY:Eva Miranda (Universitat Politècnica de Catalunya)
DTSTART:20211221T163000Z
DTEND:20211221T173000Z
DTSTAMP:20260423T022725Z
UID:Geolis/70
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Geolis/70/">
 Looking at the Euler flows through a contact mirror</a>\nby Eva Miranda (U
 niversitat Politècnica de Catalunya) as part of Geometria em Lisboa (IST)
 \n\n\nAbstract\nThe dynamics of an inviscid and incompressible fluid flow 
 on a Riemannian manifold is governed by the Euler equations. Recently\, Ta
 o [6\, 7\, 8] launched a programme to address the global existence problem
  for the Euler and Navier-Stokes equations based on the concept of univers
 ality. Inspired by this proposal\, we show that the stationary Euler equat
 ions exhibit several universality features\, in the sense that\, any non-a
 utonomous flow on a compact manifold can be extended to a smooth stationar
 y solution of the Euler equations on some Riemannian manifold of possibly 
 higher dimension [1].\n\nA key point in the proof is looking at the h-prin
 ciple in contact geometry through a contact mirror\, unveiled by Etnyre an
 d Ghrist in [4] more than two decades ago. We end this talk addressing a q
 uestion raised by Moore in [5] : “Is hydrodynamics capable of performing
  computations?”. The universality result above yields the Turing complet
 eness of the steady Euler flows on a 17-dimensional sphere. Can this resul
 t be improved? In [2] we construct a Turing complete steady Euler flow in 
 dimension 3. Time permitting\, we discuss this and other generalizations f
 or t-dependent Euler flows contained in [3].\n\nIn all the constructions a
 bove\, the metric is seen as an additional "variable" and thus the method 
 of proof does not work if the metric is prescribed.\n\nIs it still possibl
 e to construct a Turing complete Euler flow on a 3-dimensional space with 
 the standard metric? Yes\, see our recent preprint https://arxiv.org/abs/2
 111.03559 (joint with Cardona and Peralta).\n\nThis talk is based on sever
 al joint works with Cardona\, Peralta-Salas and Presas.\n\n[1] R. Cardona\
 , E. Miranda\, D. Peralta-Salas\, F. Presas. Universality of Euler flows a
 nd flexibility of Reeb embeddings\, arXiv:1911.01963.\n\n[2] R. Cardona\, 
 E. Miranda\, D. Peralta-Salas\, F. Presas. Constructing Turing complete Eu
 ler flows in dimension 3. PNAS May 11\, 2021 118 (19) e2026818118\; https:
 //doi.org/10.1073/pnas.2026818118.\n\n[3] R. Cardona\, E. Miranda and D. P
 eralta-Salas\, Turing universality of the incompressible Euler equations a
 nd a conjecture of Moore\, International Mathematics Research Notices\, rn
 ab233\, https://doi.org/10.1093/imrn/rnab233\n\n[4] J. Etnyre\, R. Ghrist.
  Contact topology and hydrodynamics I. Beltrami fields and the Seifert con
 jecture. Nonlinearity 13 (2000) 441–458.\n\n[5] C. Moore. Generalized sh
 ifts: unpredictability and undecidability in dynamical systems. Nonlineari
 ty 4 (1991) 199–230.\n\n[6] T. Tao. On the universality of potential wel
 l dynamics. Dyn. PDE 14 (2017) 219–238.\n\n[7] T. Tao. On the universali
 ty of the incompressible Euler equation on compact manifolds. Discrete Con
 t. Dyn. Sys. A 38 (2018) 1553–1565.\n\n[8] T. Tao. Searching for singula
 rities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–41
 9.\n
LOCATION:https://researchseminars.org/talk/Geolis/70/
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