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      On the universality of the incompressible Euler equation on compact manifolds

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          Abstract

          The incompressible Euler equations on a compact Riemannian manifold \((M,g)\) take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0. \end{align*} We show that any quadratic ODE \(\partial_t y = B(y,y)\), where \(B : {\bf R}^n \times {\bf R}^n \to {\bf R}^n\) is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold \(M\) if and only if \(B\) obeys the cancellation condition \(\langle B(y,y), y \rangle = 0\) for some positive definite inner product \(\langle,\rangle\) on \( {\bf R}^n\). This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.

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          Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits

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            Blowup in a three-dimensional vector model for the Euler equations

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              Finite time blowup for an averaged three-dimensional Navier-Stokes equation

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                Author and article information

                Journal
                24 July 2017
                Article
                1707.07807
                e0a3975e-74fd-4e43-99b4-c1ea0157370e

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                35Q35, 37N10, 76B99
                14 pages, no figures, submitted, Discrete and Continuous Dynamical Systems
                math.AP math.DS

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