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

On Euler's equation and 'EPDiff'

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.
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

[10]

Springer-Verlag, Berlin, 1983, Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[20]

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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