BIMSA Integrable Systems Seminar | (Higher) Monge—Ampère Geometry of the Navier—Stokes Equations

Abstract

The Poisson equation for the pressure of a homogeneous, incompressible Navier--Stokes flow is a key diagnostic relation for understanding the formation of vortices in turbulence. Building on the observation that, in two dimensions, the aforementioned equation is a Ampère equation for the stream function, this talk introduces a framework for studying this relation from the perspective of (multi-)symplectic geometry. While reviewing the geometry of Monge--Ampère equations presented by Rubtsov, D'Onofrio, and Roulstone in earlier seminars of this series, we demonstrate how an associated metric on the phase space of a two-dimensional fluid flow encodes the dominance of vorticity and strain. We then discuss how multi-symplectic geometry may be used to generalise to fluid flows on Riemannian manifolds in higher dimensions, culminating in a Weiss--Okubo-type criterion in these cases. Throughout, we make comments on how the signatures and curvatures of our structures may be interpreted in terms of the geometric and topological properties of vortices.