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
References:
[1]

"Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", Edited by Milton Abramowitz and Irene A. Stegun,, Reprint of the 1972 edition. Dover Publications, (1972).   Google Scholar

[2]

V. I. Arnold, Sur la géomtrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Annales de L'Institut Fourier, 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[3]

M. Bauer, P. Harms, and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, SIAM Journal on Imaging Sciences, 5 (2012), 244.  doi: 10.1137/100807983.  Google Scholar

[4]

Thomas Buttke, The fast adaptive vortex method,, Journal of Computational Physics, 93 (1991).  doi: 10.1016/0021-9991(91)90198-T.  Google Scholar

[5]

Roberto Camassa and Darryl Holm, An integrable shallow water equation with peaked solutions,, Physical Review Letters, 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

Alexandre Chorin, "Vorticity and Turbulence,", Springer-Verlag, (1994).   Google Scholar

[7]

Ricardo Cortez, On the accuracy of impulse methods for fluid flow,, SIAM Journal on Scientific Computing, 19 (1998), 1290.  doi: 10.1137/S1064827595293570.  Google Scholar

[8]

Darryl Holm, Jerrold Marsden and Tudor Ratiu, The Euler-Poincarè equations and semidirect products with applications to continuum theories,, Advances in Mathematics, 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[9]

Darryl Holm and Jerrold Marsden, Momentum maps and measure-valued solutions for the EPDiff equation,, in, 232 (2004), 203.  doi: 10.1007/0-8176-4419-9_8.  Google Scholar

[10]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I,", Springer-Verlag, (1983).   Google Scholar

[11]

Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 27.   Google Scholar

[12]

Mario Micheli, Peter Michor and David Mumford, Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks,, SIAM Journal on Imaging Sciences, 5 (2012), 394.  doi: 10.1137/10081678X.  Google Scholar

[13]

Mario Micheli, Peter W. Michor and David Mumford, Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds,, Izvestiya: Mathematics, 77 (2013), 541.  doi: 10.1070/IM2013v077n03ABEH002648.  Google Scholar

[14]

Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Documenta Mathematica, 10 (2005), 217.   Google Scholar

[15]

Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Applied and Computational Harmonic Analysis, 23 (2007), 74.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[16]

Peter W. Michor and David Mumford, A zoo of diffeomorphism groups on $\mathbbR^n$,, Annals of Global Ananlysis and Geometry, (2013).  doi: 10.1007/s10455-013-9380-2.  Google Scholar

[17]

Michael I. Miller, Gary E. Christensen, Yali Amit and Ulf Grenander, Mathematical textbook of deformable neuroanatomies,, Proceedings National Academy of Science, 90 (1993), 11944.  doi: 10.1073/pnas.90.24.11944.  Google Scholar

[18]

Michael Miller, Alain Trouvé and Laurent Younes, On the metrics and Euler-Lagrange equations of computational anatomy,, Annual Review of Biomedical Engineering, (2002), 375.   Google Scholar

[19]

V. I. Oseledets, On a new way of writing the Navier-Stokes equations: The Hamiltonian formalism,, Communications of the Moscow Mathematical Society (1988). Translation in Russian Mathematics Surveys, 44 (1988), 210.  doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar

[20]

P. H. Roberts, A Hamiltonian theory for weakly interacting vortices,, Mathematika, 19 (1972), 169.  doi: 10.1112/S0025579300005611.  Google Scholar

[21]

Michael E. Taylor, "Partial Differential Equations III: Nonlinear Equations,", Springer, (2010).   Google Scholar

[22]

Alain Trouvé and Laurent Younes, Local geometry of deformable templates,, SIAM Journal on Mathematical Analysis, 37 (2005), 17.  doi: 10.1137/S0036141002404838.  Google Scholar

[23]

L. Younes, "Shapes and Diffeomorphisms,", Springer, (2010).  doi: 10.1007/978-3-642-12055-8.  Google Scholar

show all references

References:
[1]

"Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", Edited by Milton Abramowitz and Irene A. Stegun,, Reprint of the 1972 edition. Dover Publications, (1972).   Google Scholar

[2]

V. I. Arnold, Sur la géomtrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Annales de L'Institut Fourier, 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[3]

M. Bauer, P. Harms, and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, SIAM Journal on Imaging Sciences, 5 (2012), 244.  doi: 10.1137/100807983.  Google Scholar

[4]

Thomas Buttke, The fast adaptive vortex method,, Journal of Computational Physics, 93 (1991).  doi: 10.1016/0021-9991(91)90198-T.  Google Scholar

[5]

Roberto Camassa and Darryl Holm, An integrable shallow water equation with peaked solutions,, Physical Review Letters, 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

Alexandre Chorin, "Vorticity and Turbulence,", Springer-Verlag, (1994).   Google Scholar

[7]

Ricardo Cortez, On the accuracy of impulse methods for fluid flow,, SIAM Journal on Scientific Computing, 19 (1998), 1290.  doi: 10.1137/S1064827595293570.  Google Scholar

[8]

Darryl Holm, Jerrold Marsden and Tudor Ratiu, The Euler-Poincarè equations and semidirect products with applications to continuum theories,, Advances in Mathematics, 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[9]

Darryl Holm and Jerrold Marsden, Momentum maps and measure-valued solutions for the EPDiff equation,, in, 232 (2004), 203.  doi: 10.1007/0-8176-4419-9_8.  Google Scholar

[10]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I,", Springer-Verlag, (1983).   Google Scholar

[11]

Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 27.   Google Scholar

[12]

Mario Micheli, Peter Michor and David Mumford, Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks,, SIAM Journal on Imaging Sciences, 5 (2012), 394.  doi: 10.1137/10081678X.  Google Scholar

[13]

Mario Micheli, Peter W. Michor and David Mumford, Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds,, Izvestiya: Mathematics, 77 (2013), 541.  doi: 10.1070/IM2013v077n03ABEH002648.  Google Scholar

[14]

Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Documenta Mathematica, 10 (2005), 217.   Google Scholar

[15]

Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Applied and Computational Harmonic Analysis, 23 (2007), 74.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[16]

Peter W. Michor and David Mumford, A zoo of diffeomorphism groups on $\mathbbR^n$,, Annals of Global Ananlysis and Geometry, (2013).  doi: 10.1007/s10455-013-9380-2.  Google Scholar

[17]

Michael I. Miller, Gary E. Christensen, Yali Amit and Ulf Grenander, Mathematical textbook of deformable neuroanatomies,, Proceedings National Academy of Science, 90 (1993), 11944.  doi: 10.1073/pnas.90.24.11944.  Google Scholar

[18]

Michael Miller, Alain Trouvé and Laurent Younes, On the metrics and Euler-Lagrange equations of computational anatomy,, Annual Review of Biomedical Engineering, (2002), 375.   Google Scholar

[19]

V. I. Oseledets, On a new way of writing the Navier-Stokes equations: The Hamiltonian formalism,, Communications of the Moscow Mathematical Society (1988). Translation in Russian Mathematics Surveys, 44 (1988), 210.  doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar

[20]

P. H. Roberts, A Hamiltonian theory for weakly interacting vortices,, Mathematika, 19 (1972), 169.  doi: 10.1112/S0025579300005611.  Google Scholar

[21]

Michael E. Taylor, "Partial Differential Equations III: Nonlinear Equations,", Springer, (2010).   Google Scholar

[22]

Alain Trouvé and Laurent Younes, Local geometry of deformable templates,, SIAM Journal on Mathematical Analysis, 37 (2005), 17.  doi: 10.1137/S0036141002404838.  Google Scholar

[23]

L. Younes, "Shapes and Diffeomorphisms,", Springer, (2010).  doi: 10.1007/978-3-642-12055-8.  Google Scholar

[1]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233

[2]

Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065

[3]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[4]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605

[5]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127

[6]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015

[7]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415

[8]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[9]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015

[10]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404

[11]

Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014

[12]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002

[13]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[14]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[15]

Chiun-Chuan Chen, Yuan Lou, Hirokazu Ninomiya, Peter Polacik, Xuefeng Wang. Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : ⅰ-ⅱ. doi: 10.3934/dcds.2020171

[16]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[17]

Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051

[18]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

[19]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027

[20]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]