# American Institute of Mathematical Sciences

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, Inc., New York, 1992.  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), 319-361. 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), 244-310. doi: 10.1137/100807983.  Google Scholar [4] Thomas Buttke, The fast adaptive vortex method, Journal of Computational Physics, 93 (1991), 485. 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), 1661-1664. 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), 1290-1302. 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), 1-81. 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 "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] Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer-Verlag, Berlin, 1983, Google Scholar [11] Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Springer Lecture Notes in Math.," 448 (1975), 27-50.  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), 394-433. 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), 541-570. 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), 217-245.  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), 74-113. 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), 11944-11948. 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), 375-405. 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 (1989), 210-211. doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar [20] P. H. Roberts, A Hamiltonian theory for weakly interacting vortices, Mathematika, 19 (1972), 169-179. 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), 17-59. doi: 10.1137/S0036141002404838.  Google Scholar [23] L. Younes, "Shapes and Diffeomorphisms," Springer, 171, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar

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##### 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, Inc., New York, 1992.  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), 319-361. 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), 244-310. doi: 10.1137/100807983.  Google Scholar [4] Thomas Buttke, The fast adaptive vortex method, Journal of Computational Physics, 93 (1991), 485. 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), 1661-1664. 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), 1290-1302. 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), 1-81. 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 "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] Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer-Verlag, Berlin, 1983, Google Scholar [11] Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Springer Lecture Notes in Math.," 448 (1975), 27-50.  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), 394-433. 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), 541-570. 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), 217-245.  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), 74-113. 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), 11944-11948. 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), 375-405. 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 (1989), 210-211. doi: 10.1070/RM1989v044n03ABEH002122.  Google Scholar [20] P. H. Roberts, A Hamiltonian theory for weakly interacting vortices, Mathematika, 19 (1972), 169-179. 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), 17-59. doi: 10.1137/S0036141002404838.  Google Scholar [23] L. Younes, "Shapes and Diffeomorphisms," Springer, 171, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar
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