Seminar talk, 28 October 2015

From Geometry of Differential Equations
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Speaker: Vladimir Rubtsov

Title: Monge-Ampère structures and the geometry of incompressible flows

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Ampère 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-Ampère structure for incompressible flow in two dimensions is shown to be hypersymplectic.

Joint work with Bertrand Banos and Ian Roulstone