A nonlinear analysis of the averaged euler equations
File(s)
Working paper
OA Location
Author(s)
Holm, Darryl D
Kouranbaeva, Shinar
Marsden, Jerrold E
Ratiu, Tudor
Shkoller, Steve
Type
Working Paper
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}.
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}.
Date Issued
1999-03-25
Citation
1999
Copyright Statement
© 1999 The Authors.
Identifier
http://arxiv.org/abs/chao-dyn/9903036v1
Subjects
chao-dyn
chao-dyn
nlin.CD
Notes
25 pages, no figures, Dedicated to Vladimir Arnold on the occasion of his 60th birthday, Arnold Festschrift Volume 2 (in press)
Publication Status
Published