American Institute of Mathematical Sciences

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June  2017, 9(2): 167-189. doi: 10.3934/jgm.2017007

Local well-posedness of the EPDiff equation: A survey

Boris Kolev

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

Received  December 2015 Revised  August 2016 Published  May 2017

This article is a survey on the local well-posedness problem for the general EPDiff equation. The main contribution concerns recent results on local existence of the geodesics on $\text{Dif}{{\text{f}}^{\infty }}\left( {{\mathbb{T}}^{d}} \right)$ and $\text{Dif}{{\text{f}}^{\infty }}\left( {{\mathbb{R}}^{d}} \right)$ when the inertia operator is a non-local Fourier multiplier.

Keywords: EPDiff equation, diffeomorphism groups, Sobolev metrics of fractional order.
Mathematics Subject Classification: Primary: 58D05; Secondary: 35Q35.
Citation: Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[2]

V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, vol. 125 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998.  Google Scholar

[3]

V. I. Averbukh and O. G. Smolyanov, The various definitions of the derivative in linear topological spaces, Russian Math. Surveys, 23 (1968), 67-116.   Google Scholar

[4]

M. Bauer, J. Escher and B. Kolev, Local and Global Well-posedness of the fractional order EPDiff equation on ${R}^d$, Journal of Differential Equations, 258 (2015), 2010-2053.  doi: 10.1016/j.jde.2014.11.021.  Google Scholar

[5]

M. Bauer, B. Kolev and S. C. Preston, Geometric investigations of a vorticity model equation, J. Differential Equations, 260 (2016), 478-516.  doi: 10.1016/j.jde.2015.09.030.  Google Scholar

[6]

Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc., 2 (1989), 225-255.  doi: 10.1090/S0894-0347-1989-0969419-8.  Google Scholar

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Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), 411-452.  doi: 10.1002/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3.  Google Scholar

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R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

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J. -Y. Chemin, Équations d'Euler d'un fluide incompressible, in Facettes mathématiques de la mécanique des fluides, Ed. Éc. Polytech., Palaiseau, 2010, 9-30. Google Scholar

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E. Cismas, Euler-Poincaré-Arnold equations on semi-direct products, Discrete and Continuous Dynamical Systems -Series A, 36 (2016), 5993-6022.  doi: 10.3934/dcds.2016063.  Google Scholar

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A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.  doi: 10.1007/s10455-006-9042-8.  Google Scholar

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A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

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J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

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J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968.  doi: 10.1007/s00028-014-0245-3.  Google Scholar

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J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Commun. Pure Appl. Anal., 11 (2012), 1407-1419.  doi: 10.3934/cpaa.2012.11.1407.  Google Scholar

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A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1988, A Wiley-Interscience Publication.  Google Scholar

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show all references

References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[2]

V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, vol. 125 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998.  Google Scholar

[3]

V. I. Averbukh and O. G. Smolyanov, The various definitions of the derivative in linear topological spaces, Russian Math. Surveys, 23 (1968), 67-116.   Google Scholar

[4]

M. Bauer, J. Escher and B. Kolev, Local and Global Well-posedness of the fractional order EPDiff equation on ${R}^d$, Journal of Differential Equations, 258 (2015), 2010-2053.  doi: 10.1016/j.jde.2014.11.021.  Google Scholar

[5]

M. Bauer, B. Kolev and S. C. Preston, Geometric investigations of a vorticity model equation, J. Differential Equations, 260 (2016), 478-516.  doi: 10.1016/j.jde.2015.09.030.  Google Scholar

[6]

Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc., 2 (1989), 225-255.  doi: 10.1090/S0894-0347-1989-0969419-8.  Google Scholar

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Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), 411-452.  doi: 10.1002/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3.  Google Scholar

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[9]

J. -Y. Chemin, Équations d'Euler d'un fluide incompressible, in Facettes mathématiques de la mécanique des fluides, Ed. Éc. Polytech., Palaiseau, 2010, 9-30. Google Scholar

[10]

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[11]

A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.  doi: 10.1007/s10455-006-9042-8.  Google Scholar

[12]

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[13]

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[14]

P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724.  doi: 10.1002/cpa.3160380605.  Google Scholar

[15]

A. Degasperis, D. D. Holm and A. N. I. Hone, A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 170-183.  doi: 10.1023/A:1021186408422.  Google Scholar

[16]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 1999, 23-37.  Google Scholar

[17]

D. G. Ebin, J. E. Marsden and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, in Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress Differential Geometry and Applications, (Dalhousie Univ., Halifax, N. S., 1971), Canad. Math. Congr., Montreal, Que., 1 (1972), 135-279.  Google Scholar

[18]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.(2), 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[19]

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[20]

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[21]

D. G. Ebin, A concise presentation of the Euler equations of hydrodynamics, Comm. Partial Differential Equations, 9 (1984), 539-559.  doi: 10.1080/03605308408820341.  Google Scholar

[22]

H. I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194.  doi: 10.4310/jdg/1214427887.  Google Scholar

[23]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

[24]

J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968.  doi: 10.1007/s00028-014-0245-3.  Google Scholar

[25]

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372.  doi: 10.3934/jgm.2014.6.335.  Google Scholar

[26]

J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Commun. Pure Appl. Anal., 11 (2012), 1407-1419.  doi: 10.3934/cpaa.2012.11.1407.  Google Scholar

[27]

L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154-193.   Google Scholar

[28]

A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1988, A Wiley-Interscience Publication.  Google Scholar

[29]

F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009. Google Scholar

[30]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braş sov Ser. Ⅲ, 2 (2009), 55-58.   Google Scholar

[31]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[32]

N. Hermas and S. Djebali, Existence de l'application exponentielle riemannienne d'un groupe de difféomorphismes muni d'une métrique de Sobolev, J. Math. Pures Appl.(9), 94 (2010), 433-446.  doi: 10.1016/j.matpur.2009.11.004.  Google Scholar

[33]

P. Iglesias-Zemmour, Diffeology, vol. 185 of Mathematical Surveys and Monographs, American Mathematical Society, 2013. doi: 10.1090/surv/185.  Google Scholar

[34]

H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society, 1st edition, American Mathematical Society, 2013. doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[35]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[36]

H. H. Keller, Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics, Vol. 417, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[37]

B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144.  doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[38]

B. Khesin and V. Ovsienko, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.   Google Scholar

[39]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.  doi: 10.1063/1.532690.  Google Scholar

[40]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, vol. 53 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/053.  Google Scholar

[41]

S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[42]

J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064.  doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[43]

J. Lenells, The Hunter-Saxton equation: A geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277.  doi: 10.1137/050647451.  Google Scholar

[44]

J. Lenells, G. Misiołek and F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[45]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  Google Scholar

[46]

R. McLachlan and X. Zhang, Well-posedness of modified Camassa-Holm equations, J. Differential Equations, 246 (2009), 3241-3259.  doi: 10.1016/j.jde.2009.01.039.  Google Scholar

[47]

P. W. Michor, Manifolds of Differentiable Mappings, vol. 3 of Shiva Mathematics Series, Shiva Publishing Ltd., Nantwich, 1980.  Google Scholar

[48]

P. W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, in Phase Space Analysis of Partial Differential Equations, vol. 69 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2006,133-215. doi: 10.1007/978-0-8176-4521-2_11.  Google Scholar

[49]

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