On a Lagrangian formulation of the incompressible Euler equation

TL;DR

This paper demonstrates that the incompressible Euler equations can be formulated as a geodesic problem on an infinite-dimensional manifold of volume-preserving diffeomorphisms, with an analytic Christoffel map.

Contribution

It provides a Lagrangian coordinate formulation of the Euler equations as a geodesic flow, with an analytic geometric structure on the diffeomorphism group.

Findings

Euler equations expressed as geodesic equations on a manifold

Christoffel map is real analytic

Sobolev class vector fields generate curves on the diffeomorphism group

Abstract

In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\R^n)$, $s > n/2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.