# 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. [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. [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. [4] Thomas Buttke, The fast adaptive vortex method, Journal of Computational Physics, 93 (1991), 485. 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), 1661-1664. 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), 1290-1302. 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), 1-81. doi: 10.1006/aima.1998.1721. [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. [10] Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer-Verlag, Berlin, 1983, [11] Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Springer Lecture Notes in Math.," 448 (1975), 27-50. [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. [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. [14] Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Documenta Mathematica, 10 (2005), 217-245. [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. [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), 11944-11948. 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), 375-405. [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. [20] P. H. Roberts, A Hamiltonian theory for weakly interacting vortices, Mathematika, 19 (1972), 169-179. 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), 17-59. doi: 10.1137/S0036141002404838. [23] L. Younes, "Shapes and Diffeomorphisms," Springer, 171, 2010. doi: 10.1007/978-3-642-12055-8.

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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. [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. [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. [4] Thomas Buttke, The fast adaptive vortex method, Journal of Computational Physics, 93 (1991), 485. 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), 1661-1664. 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), 1290-1302. 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), 1-81. doi: 10.1006/aima.1998.1721. [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. [10] Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer-Verlag, Berlin, 1983, [11] Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Springer Lecture Notes in Math.," 448 (1975), 27-50. [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. [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. [14] Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Documenta Mathematica, 10 (2005), 217-245. [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. [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), 11944-11948. 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), 375-405. [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. [20] P. H. Roberts, A Hamiltonian theory for weakly interacting vortices, Mathematika, 19 (1972), 169-179. 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), 17-59. doi: 10.1137/S0036141002404838. [23] L. Younes, "Shapes and Diffeomorphisms," Springer, 171, 2010. doi: 10.1007/978-3-642-12055-8.
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