The Euler equations describe the dynamics of an inviscid and incompressible fluid on a Riemannian manifold. In this context, Arnold's structure theorem marked the birth of the modern field of Topological Hydrodynamics. This theorem provides an almost complete description of the rigid behavior of a solution whose Bernoulli function is analytic or Morse-Bott on a three dimensional manifold. However, only a few examples of such fluids exist in the literature. We will address the inverse problem to Arnold's theorem, proving a realization and topological classification theorem for non-vanishing Eulerisable flows (steady solutions for some metric) with a Morse-Bott Bernoulli function. The proofs combine the geometry of the equations for a varying metric with the theory of Hamiltonian integrable systems. Lastly, we drop the non-vanishing assumption and investigate how the topology of the ambient manifold can be an obstruction to the existence of any Bott integrable fluid for any metric.
The talk will be via Zoom at: