American Institute of Mathematical Sciences

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September  2013, 5(3): 319-344. doi: 10.3934/jgm.2013.5.319

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, United States
2. 

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria

Received  November 2012 Revised  June 2013 Published  September 2013

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.
Keywords: Diffeomorphism groups, vortex-solitons, vortons., EPDiff, Euler's equation.
Mathematics Subject Classification: Primary: 35Q31, 58B20, 58D0.
Citation: David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
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show all references

References:
[1]

Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992.  Google Scholar

[2]

Annales de L'Institut Fourier, 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[3]

SIAM Journal on Imaging Sciences, 5 (2012), 244-310. doi: 10.1137/100807983.  Google Scholar

[4]

Journal of Computational Physics, 93 (1991), 485. doi: 10.1016/0021-9991(91)90198-T.  Google Scholar

[5]

Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

Springer-Verlag, 1994.  Google Scholar

[7]

SIAM Journal on Scientific Computing, 19 (1998), 1290-1302. doi: 10.1137/S1064827595293570.  Google Scholar

[8]

Advances in Mathematics, 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[9]

in "The Breadth of Symplectic and Poisson Geometry, A festschrift for Alan Weinstein," Progress in Mathematics, 232 (2004), 203-235. doi: 10.1007/0-8176-4419-9_8.  Google Scholar

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Springer-Verlag, Berlin, 1983, Google Scholar

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in "Springer Lecture Notes in Math.," 448 (1975), 27-50.  Google Scholar

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SIAM Journal on Imaging Sciences, 5 (2012), 394-433. doi: 10.1137/10081678X.  Google Scholar

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Izvestiya: Mathematics, 77 (2013), 541-570. doi: 10.1070/IM2013v077n03ABEH002648.  Google Scholar

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Documenta Mathematica, 10 (2005), 217-245.  Google Scholar

[15]

Applied and Computational Harmonic Analysis, 23 (2007), 74-113. doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[16]

Annals of Global Ananlysis and Geometry, (2013). doi: 10.1007/s10455-013-9380-2.  Google Scholar

[17]

Proceedings National Academy of Science, 90 (1993), 11944-11948. doi: 10.1073/pnas.90.24.11944.  Google Scholar

[18]

Annual Review of Biomedical Engineering, (2002), 375-405. Google Scholar

[19]

Communications of the Moscow Mathematical Society (1988). Translation in Russian Mathematics Surveys, 44 (1989), 210-211. doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar

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Mathematika, 19 (1972), 169-179. doi: 10.1112/S0025579300005611.  Google Scholar

[21]

Springer, 2010.  Google Scholar

[22]

SIAM Journal on Mathematical Analysis, 37 (2005), 17-59. doi: 10.1137/S0036141002404838.  Google Scholar

[23]

Springer, 171, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar

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