Monge–Ampère geometry and vortices
Résumé
We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of- strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associ- ated (higher) Lagrangian submanifold in the cotangent bundle to the config- uration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill’s spherical vortex and an integrable case of Arnol’d–Beltrami–Childress flow, have symmetries that facilitate a formu- lation of these solutions from the perspective of (higher) symplectic reduction.
Mots clés
Monge-Ampere equations Navier-Stokes equations Lagrangian submanifolds symplectic reduction (pseudo-) Riemannian geometry higher symplectic geometry vortex topology Mathematics Subject Classification numbers: 76D05 53B30 53B35 53D05 53D20 53D12 3 2. Connection C.2.3. Curvature References
Monge-Ampere equations
Navier-Stokes equations
Lagrangian submanifolds
symplectic reduction
(pseudo-) Riemannian geometry
higher symplectic geometry
vortex topology Mathematics Subject Classification numbers: 76D05
53B30
53B35
53D05
53D20
53D12 3 2. Connection C.2.3. Curvature References
Origine : Fichiers éditeurs autorisés sur une archive ouverte