K"ahler Geometry and the Navier-Stokes Equations

Ian Roulstone; Bertrand Banos; John D. Gibbon; Vladimir Roubtsov

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arXiv · nlin/0509023

K"ahler Geometry and the Navier-Stokes Equations

Ian Roulstone·Bertrand Banos·John D. Gibbon·Vladimir Roubtsov

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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--Ampère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a Kähler geometry is described, which is associated with the Monge--Ampère problem. This Kähler 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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BibTeX RIS

Ian Roulstone, Bertrand Banos, John D. Gibbon, Vladimir Roubtsov. 2005-09-09. K"ahler Geometry and the Navier-Stokes Equations. https://arxiv.org/abs/nlin/0509023

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