K"ahler Geometry and the Navier-Stokes Equations

TL;DR

This paper explores the geometric structures underlying incompressible fluid dynamics, revealing connections to Kähler and Calabi--Yau geometries in 2D and 3D flows, and generalizing these structures to complex flow scenarios.

Contribution

It introduces a novel geometric framework linking Monge--Ampère equations to Kähler and Calabi--Yau structures in fluid dynamics, extending understanding of flow geometries.

Findings

Kähler geometry describes 2D incompressible flows.

Generalized Calabi--Yau structures appear in 3D flows.

Flow constraints lead to Monge--Ampère type 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\`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.