Kahler Geometry and Burgers' Vortices

Abstract

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation in two spatial dimensions using Monge-Ampere structures. In two dimensional flows where the Laplacian of the pressure is positive, a Kahler 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-Ampere equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions.