Séminaires > Séminaire Jeudi 26 Sept. 2024
Monge-Ampère PDEs and their Geometry : A Strange Relationship
Lewis Napper
University of Surrey
Date : Jeudi 26 septembre à 13h
Lieu : Salle séminaire
Résumé :
Monge–Ampère equations (MAEs) are a class of quasi-linear, second order partial differential equations, which underlie many models of physically interesting systems.
These include : the Khokhlov–Zabolotskaya equation for three-dimensional acoustics, the Poisson equation for the pressure of an incompressible fluid, the reaction-diffusion equation, and even the Laplace and wave equations. The first part of this talk is intended as an introduction to the geometry of MAEs as a tool for studying physical systems. We begin by defining the family of MAEs in m variables and explain how they can be encoded as constraints on m-dimensional submanifolds of some 2m-dimensional phase space. Specialising to two variables, we show how the properties of these submanifolds can be used to derive properties of the solutions to a given MAE, via the Lychagin–Rubtsov theorem and metric [1]. Finally, we provide an example, applying these tools to the Poisson equation for the pressure of an incompressible fluid flow to produce a relationship between the signature of the Lychagin–Rubtsov metric and the dominance of vorticity and strain in the fluid. While some basic knowledge of differential geometry will be useful, I will not assume much a-priori and this portion of the talk will be relatively self-contained.
The second part of this talk will cover recent progress in the field of Monge–Ampère geometry, arising from further mathematical investigation of the Poisson equation. In particular, a covariant re-formulation allows our results to be applied to fluids on non-Euclidean backgrounds, e.g. on a sphere. Furthermore, we show that the correct choice of symplectic structure on the phase space can be used to couple the Poisson equation to the incompressibility constraint and that this construction generalises to higher dimensions if we take a multi-symplectic view. This leads to the open problem of defining a class of multi-symplectic Monge–Ampère equations [2].
[1] A. Kushner, V. Lychagin, and V. Rubtsov, Contact geometry and non-linear differential equations, Cambridge University Press, 2007.
[2] L. Napper, I. Roulstone, V. Rubtsov, and M. Wolf, Monge–Ampère geometry and vortices, Nonlinearity 37 (2024) 045012 [2302.11604v2].