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Communication Dans Un Congrès Année : 2009

Kahler geometry and Burgers' vortices

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Résumé

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.

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Dates et versions

hal-03031588 , version 1 (30-11-2020)

Identifiants

  • HAL Id : hal-03031588 , version 1
  • OKINA : ua65

Citer

Ian Roulstone, Bertrand Banos, J. Gibbon, Vladimir Roubtsov. Kahler geometry and Burgers' vortices. Ukrainian Mathematical Congress, Aug 2009, Kiev, Ukraine. pp.303 - 321. ⟨hal-03031588⟩
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