# 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, (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

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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, (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
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