(Higher) Monge—Ampere 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 Monge--Amp{\`e}re 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--Amp{\`e}re 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.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( paper | slides | video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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