On Euler's equation and 'EPDiff'

David Mumford; Peter W. Michor

doi:10.3934/jgm.2013.5.319

Article Contents

2013, Volume 5, Issue 3: 319-344. Doi: 10.3934/jgm.2013.5.319

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On Euler's equation and 'EPDiff'

David Mumford^1, and
Peter W. Michor^2,

1.
Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912
2.
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien

Received: October 31, 2012

Revised: May 31, 2013

Published: August 2013

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Abstract

We study a family of approximations to Euler's equation depending on two parameters $\epsilon,η \ge 0$. When $\epsilon = η = 0$ 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 ${Diff}_{H^\infty}(\mathbb{R}^n)$ or, if $\epsilon = 0$, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by $$ ||v||_{\epsilon,η} = \int_{\mathbb{R}^n} \langle L_{\epsilon,η} v, v \rangle\, dx $$ where $L_{\epsilon,η} = (I-\frac{η^2}{p} \triangle)^p \circ (I-\frac {1}{\epsilon^2} \nabla \circ div)$. All geodesic equations are locally well-posed, and the $L_{\epsilon,η}$-equation admits solutions for all time if $η > 0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to ``vortex-solitons", also called ``landmarks" in imaging science, and to new numeric approximations to fluids.

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Mathematics Subject Classification: Primary: 35Q31, 58B20, 58D05.

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