Â鶹ÊÓƵ

Abstract

For partial differential equations of Monge-Ampère type, it has been shown [1, 2] that solutions correspond to Lagrangian submanifolds of the associated phase space. Building from the observation that the Poisson equation for the pressure of an incompressible, two-dimensional, Navier-Stokes flow is a Monge-Ampère equation, this talk introduces a framework for studying fluid dynamics using properties of the aforementioned submanifolds. In particular, it is noted that such a submanifold may be equipped with a metric whose signature acts as a diagnostic for the dominance of vorticity and strain. We provide an illustrative example in two dimensions and probe the motivational question [3,4]: ‘What is a vortex?’ We conclude with comments on extensions to fluid flows in higher dimensions and some open questions.

References

1. V. V. Lychagin, V. N. Rubtsov, and I. V. Chekalov, A classification of Monge-Ampère equations, .
2. A. Kushner, V. Lychagin, and V. Rubtsov, Contact geometry and non-linear differential equations, Cambridge University Press, 2007.
3. M. Larchevêque, Pressure field, vorticity field, and coherent structures in two-dimensional incompressible turbulent flows, .
4. J. D. Gibbon, A. S. Fokas, and C. R. Doering, Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations, .