Fetching the paper…

Reading the bibliography…

2012

On Euler's equation and `EPDiff'

Mumford, David, Michor, Peter W.

Original
PDF
Post on X

Understand

We study a family of approximations to Euler's equation depending on two parameters $\varepsilon,\eta \ge 0$.

When $\varepsilon=\eta=0$ we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science.
These are all geodesic equations on either the full diffeomorphism group $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$ or, if $\varepsilon = 0$, its volume preserving subgroup.
They are defined by the right invariant metric induced by the norm on vector fields given by $$ \|v\|_{\varepsilon,\eta} = \int_{\mathbb R^n} <L_{\varepsilon,\eta} v, v> dx $$ where $L_{\varepsilon,\eta} = (I-\tfrac{\eta^2}{p} \triangle)^p \circ (I-\tfrac1{\varepsilon^2} \nabla \circ \div)$.

On Euler's equation and `EPDiff' · Around

Built on

V.I. Arnold
1966
Earlier work this paper cites.
P. H. Roberts
1972
Earlier work this paper cites.
Tosio Kato
1975
Earlier work this paper cites.
Lars Hörmander,
1983
Earlier work this paper cites.
V. I. Oseledets
1989
Earlier work this paper cites.
Thomas Buttke
1991
Earlier work this paper cites.
Handbook of mathematical functions with formulas, graphs, and mathematical tables
1992
Earlier work this paper cites.
Roberto Camassa and Darryl Holm
1993
Earlier work this paper cites.

Similar

Michael I. Miller , Gary E. Christensen , Yali Amit and Ulf Grenander
1993
Cited alongside, same era.
Alexandre Chorin
1994
Cited alongside, same era.
Ricardo Cortez
1998
Cited alongside, same era.
Darryl Holm, Jerry Marsden and Tudor Ratiu
1998
Cited alongside, same era.
Michael Miller, Alain Trouvé and Laurent Younes
2002
Cited alongside, same era.
Darryl Holm and Jerry Marsden
2004
Cited alongside, same era.
Peter Michor, David Mumford
2005
Cited alongside, same era.

Then

Peter W. Michor and David Mumford
2007
Later among the works it cites.
Michael E. Taylor
2010
Later among the works it cites.
L. Younes
Original
2010
Later among the works it cites.
M. Bauer, P. Harms, and P. W. Michor
2012
Closest in time.
Mario Micheli, Peter Michor, David Mumford
Original
2012
Closest in time.
Mario Micheli, Peter W. Michor, David Mumford
Original
2013
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Alain Trouvé, Laurent Younes

2005

Cited alongside, same era.

Peter W. Michor and David Mumford

2013

Closest in time.