The topology of Bott integrable fluids

TL;DR

This paper constructs and classifies non-vanishing steady solutions to the Euler equations on graph manifolds, linking the topology of these manifolds to the properties of integrable fluid flows.

Contribution

It demonstrates that any admissible Morse-Bott function can serve as the Bernoulli function for a steady Euler flow on graph manifolds, providing a topological classification.

Findings

Steady Euler flows with Morse-Bott Bernoulli functions exist only on graph manifolds.

Any admissible Morse-Bott function can be realized as a Bernoulli function in such flows.

Topological obstructions prevent such flows on non-graph manifolds.

Abstract

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.