2pm - 3pm
Monday 6 February 2023
Monge-Ampère Geometry and the Navier-Stokes Equations
A joint seminar in the Fields, Strings, and Geometry Group, and Nonlinear Waves and Geometry Fluid Dynamics Group research seminar series.
University of Surrey
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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.
- V. V. Lychagin, V. N. Rubtsov, and I. V. Chekalov, A classification of Monge-Ampère equations, Ann. Sci. Ec. Norm. Sup. 26 (1993) 281.
- A. Kushner, V. Lychagin, and V. Rubtsov, Contact geometry and non-linear differential equations, Cambridge University Press, 2007.
- M. Larchevêque, Pressure field, vorticity field, and coherent structures in two-dimensional incompressible turbulent flows, Theor. Comp. Fluid Dynamics 5 (1993) 215.
- J. D. Gibbon, A. S. Fokas, and C. R. Doering, Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations, Phys. D. Nonlin. Phen. 132 (1999) 497.