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

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] 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 [2] 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 [3] 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 [4] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [5] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [6] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [7] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [8] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [9] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [10] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [11] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [12] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [13] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [14] Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262 [15] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [16] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [17] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [18] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [19] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [20] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

2019 Impact Factor: 0.649