A Nonlinear Analysis of the Averaged Euler Equations

TL;DR

This paper analyzes the geometric and analytical properties of the averaged Euler equations, revealing their interpretation as geodesics on volume-preserving diffeomorphism groups with an $H^1$ metric, and explores their relation to second grade fluids.

Contribution

It introduces a geometric framework for the averaged Euler equations, showing they are geodesics on a diffeomorphism group with an $H^1$ metric and relates them to second grade fluids.

Findings

Averaged Euler equations are geodesics on a volume-preserving diffeomorphism group with an $H^1$ metric.

The equations are equivalent to those for a certain second grade fluid.

Zero viscosity limit is regular even with boundaries.

Abstract

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $\alpha$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $\alpha$. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold's theorem), but with respect to a right invariant $H^1$ metric instead of the $L^2$ metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {\it even in the presence of boundaries}.