K"ahler Geometry and the Navier-Stokes Equations
Exactly Solvable and Integrable Systems
2007-05-23 v1 Mathematical Physics
math.MP
Fluid Dynamics
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\`ere type for the stream function, when the Laplacian of the pressure is known. In two dimensions a K\"ahler geometry is described, which is associated with the Monge--Amp\`ere problem. This K\"ahler 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.
Keywords
Cite
@article{arxiv.nlin/0509023,
title = {K"ahler Geometry and the Navier-Stokes Equations},
author = {Ian Roulstone and Bertrand Banos and John D. Gibbon and Vladimir Roubtsov},
journal= {arXiv preprint arXiv:nlin/0509023},
year = {2007}
}
Comments
LaTeX Roy.Soc style, 16 pages