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Kähler Geometry and the Navier-Stokes Equations

Abstract : We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge--Amp\ère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a K\\ähler geometry is described, which is associated with the Monge--Amp\ère problem. This K\\ähler structure is then generalised to `two-and-a-half dimensional\' flows, of which Burgers\' vortex is one example. In three dimensions, we show how a generalized Calabi--Yau structure emerges in a special case.
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Contributor : Secrétariat Math. Angers <>
Submitted on : Tuesday, January 31, 2006 - 2:01:50 PM
Last modification on : Wednesday, April 1, 2020 - 1:57:20 AM

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Ian Roulstone, Bertrand Banos, John D. Gibbon, Vladimir Roubtsov. Kähler Geometry and the Navier-Stokes Equations. 2005. ⟨hal-00009622⟩