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doi: 10.3934/dcds.2022054
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## The topology of Bott integrable fluids

 Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France

*Corresponding author: Robert Cardona

Received  November 2021 Early access April 2022

We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds.

Citation: Robert Cardona. The topology of Bott integrable fluids. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022054
##### References:

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##### References:
Height function in $\Sigma_0$
Example of graph representation
Level sets of $h$
Non simple $2$-atom
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