Algebra, Topology and Algebraic Topology of 3D Ideal Fluids

TL;DR

This paper explores the algebraic and topological structures underlying 3D ideal fluid dynamics, proposing finite and infinite dimensional algebraic models to better understand Euler's equations and their approximations.

Contribution

It introduces a finite dimensional fluid algebra model and discusses how Lie infinity algebra structures can approximate the classical infinite dimensional fluid algebra.

Findings

Finite dimensional fluid algebra models capture key properties of fluid motion.

Lie infinity algebra structures can approximate the Jacobi identity in finite models.

Potential for new algorithms based on algebraic structures for fluid simulation.

Abstract

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimensional 3D fluid algebra", write its Euler equation and derive properties related to energy, helicity, transport of vorticity and linking that characterize this equation. This is directly motivated by the infinite dimensional fluid algebra associated to a closed riemannian three manifold whose Euler equation as defined above is the Euler PDE of fluid motion. The classical infinite dimensional fluid algebra satisfies an additional identity related to the Jacobi identity for the lie bracket of vector fields. In part II we discuss informally how this Jacobi identity can be reestablished in finite dimensional approximations as a Lie infinity algebra. The main point of a developed version of this theory would be a coherence between various levels of approximation. It is hoped that a better understanding of the meaning of the Euler equation in terms of such infinity structures would yield algorithms of computation that work well for conceptual reasons.