Monge-Ampère Structures and the Geometry of Incompressible Flows

Bertrand Banos (1) , Vladimir Roubtsov (2) , Ian Roulstone (3)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
3 UNIS - University of Surrey

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere structure, and Burgers'-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction The Monge-Amp\`ere structure for incompressible flow in two dimensions is shown to be hypersymplectic.

Mots clés: Euler equations Navier-Stokes equations; Monge-Ampère operators

Domaines: Physique mathématique [math-ph]

Bertrand Banos, Vladimir Roubtsov, Ian Roulstone. Monge-Ampère Structures and the Geometry of Incompressible Flows. 2015. ⟨hal-01228550v2⟩