Kahler geometry and Burgers' vortices
Roulstone, I, Banos, B, Gibbon, JD and Roubtsov, V (2009) Kahler geometry and Burgers' vortices Proceedings of Ukrainian National Academy Mathematics, 16 (2). 303 - 321.
We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two spatial dimensions. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation using Monge-Amp`ere structures. In two dimensional flows where the laplacian of the pressure is positive, a K¨ahler geometry is described on the phase space of the fluid; in regions where the laplacian of the pressure is negative, a product structure is described. These structures can be related to the ellipticity and hyperbolicity (respectively) of a Monge-Amp`ere equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions.
|Divisions:||Faculty of Engineering and Physical Sciences > Mathematics|
|Deposited By:||Symplectic Elements|
|Deposited On:||03 Feb 2012 13:18|
|Last Modified:||24 Jan 2013 09:23|
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