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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 |
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).
|
[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. |
[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. |
[4] |
Thomas Buttke, The fast adaptive vortex method,, Journal of Computational Physics, 93 (1991).
doi: 10.1016/0021-9991(91)90198-T. |
[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. |
[6] |
Alexandre Chorin, "Vorticity and Turbulence,", Springer-Verlag, (1994).
|
[7] |
Ricardo Cortez, On the accuracy of impulse methods for fluid flow,, SIAM Journal on Scientific Computing, 19 (1998), 1290.
doi: 10.1137/S1064827595293570. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[14] |
Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Documenta Mathematica, 10 (2005), 217.
|
[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. |
[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. |
[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. |
[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. |
[20] |
P. H. Roberts, A Hamiltonian theory for weakly interacting vortices,, Mathematika, 19 (1972), 169.
doi: 10.1112/S0025579300005611. |
[21] |
Michael E. Taylor, "Partial Differential Equations III: Nonlinear Equations,", Springer, (2010).
|
[22] |
Alain Trouvé and Laurent Younes, Local geometry of deformable templates,, SIAM Journal on Mathematical Analysis, 37 (2005), 17.
doi: 10.1137/S0036141002404838. |
[23] |
L. Younes, "Shapes and Diffeomorphisms,", Springer, (2010).
doi: 10.1007/978-3-642-12055-8. |
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).
|
[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. |
[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. |
[4] |
Thomas Buttke, The fast adaptive vortex method,, Journal of Computational Physics, 93 (1991).
doi: 10.1016/0021-9991(91)90198-T. |
[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. |
[6] |
Alexandre Chorin, "Vorticity and Turbulence,", Springer-Verlag, (1994).
|
[7] |
Ricardo Cortez, On the accuracy of impulse methods for fluid flow,, SIAM Journal on Scientific Computing, 19 (1998), 1290.
doi: 10.1137/S1064827595293570. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[14] |
Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Documenta Mathematica, 10 (2005), 217.
|
[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. |
[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. |
[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. |
[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. |
[20] |
P. H. Roberts, A Hamiltonian theory for weakly interacting vortices,, Mathematika, 19 (1972), 169.
doi: 10.1112/S0025579300005611. |
[21] |
Michael E. Taylor, "Partial Differential Equations III: Nonlinear Equations,", Springer, (2010).
|
[22] |
Alain Trouvé and Laurent Younes, Local geometry of deformable templates,, SIAM Journal on Mathematical Analysis, 37 (2005), 17.
doi: 10.1137/S0036141002404838. |
[23] |
L. Younes, "Shapes and Diffeomorphisms,", Springer, (2010).
doi: 10.1007/978-3-642-12055-8. |
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