Local well-posedness of the EPDiff equation: A survey

Previous Article
Möbius invariants in image recognition
JGM Home
This Issue
Next Article
The Madelung transform as a momentum map

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

Local well-posedness of the EPDiff equation: A survey

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.
[2]	V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, vol. 125 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998.
[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.
[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.
[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.
[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.
[7]	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.
[8]	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.
[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.
[10]	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.
[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.
[12]	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.
[13]	A. Constantin and B. Kolev, On the geometry of the diffeomorphism group of the circle, in Number Theory, Analysis and Geometry, Springer, New York, 2012,143-160. doi: 10.1007/978-1-4614-1260-1_7.
[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.
[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.
[16]	A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 1999, 23-37.
[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.
[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.
[19]	D. G. Ebin, On the space of Riemannian metrics, Bull. Amer. Math. Soc., 74 (1968), 1001-1003. doi: 10.1090/S0002-9904-1968-12115-9.
[20]	D. G. Ebin, The manifold of Riemannian metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970, 11-40.
[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.
[22]	H. I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194. doi: 10.4310/jdg/1214427887.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009.
[30]	F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braş sov Ser. Ⅲ, 2 (2009), 55-58.
[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.
[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.
[33]	P. Iglesias-Zemmour, Diffeology, vol. 185 of Mathematical Surveys and Monographs, American Mathematical Society, 2013. doi: 10.1090/surv/185.
[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.
[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.
[36]	H. H. Keller, Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics, Vol. 417, Springer-Verlag, Berlin-New York, 1974.
[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.
[38]	B. Khesin and V. Ovsienko, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.
[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.
[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.
[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.
[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.
[43]	J. Lenells, The Hunter-Saxton equation: A geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. doi: 10.1137/050647451.
[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.
[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.
[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.
[47]	P. W. Michor, Manifolds of Differentiable Mappings, vol. 3 of Shiva Mathematics Series, Shiva Publishing Ltd., Nantwich, 1980.
[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.
[49]	P. Michor and D. Mumford, On Euler's equation and 'EPDiff', The Journal of Geometric Mechanics, 5 (2013), 319-344. doi: 10.3934/jgm.2013.5.319.
[50]	J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984,1007-1057.
[51]	G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[52]	G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104. doi: 10.1007/PL00012648.
[53]	G. Misiolek and S. C. Preston, Fredholm properties of Riemannian exponential maps on diffeomorphism groups, Invent. Math., 179 (2010), 191-227. doi: 10.1007/s00222-009-0217-3.
[54]	J. J. Moreau, Une méthode de "cinématique fonctionnelle" en hydrodynamique, C. R. Acad. Sci. Paris, 249 (1959), 2156-2158.
[55]	H. Omori, On the group of diffeomorphisms on a compact manifold, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970,167-183.
[56]	H. Omori, Infinite-dimensional Lie Groups, vol. 158 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author.
[57]	R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[58]	H. Poincaré, Sur une nouvelle forme des équations de la mécanique, C.R. Acad. Sci., 132 (1901), 369-371.
[59]	M. Ruzhansky and V. Turunen, Pseudo-differential Operators and Symmetries, vol. 2 of Pseudo-Differential Operators. Theory and Applications, Birkhäuser Verlag, Basel, 2010, Background analysis and advanced topics. doi: 10.1007/978-3-7643-8514-9.
[60]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335.
[61]	S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom., 55 (2000), 145-191. doi: 10.4310/jdg/1090340568.
[62]	A. Shnirelman, Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, 4 (1994), 586-620. doi: 10.1007/BF01896409.
[63]	A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128 (1985), 82-109,144.
[64]	A. Trouvé and L. Younes, Local geometry of deformable templates, SIAM J. Math. Anal., 37 (2005), 17-59 (electronic). doi: 10.1137/S0036141002404838.
[65]	M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric, J. Nonlinear Math. Phys., 17 (2010), 7-11. doi: 10.1142/S1402925110000544.

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.
[2]	V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics, vol. 125 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998.
[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.
[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.
[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.
[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.
[7]	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.
[8]	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.
[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.
[10]	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.
[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.
[12]	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.
[13]	A. Constantin and B. Kolev, On the geometry of the diffeomorphism group of the circle, in Number Theory, Analysis and Geometry, Springer, New York, 2012,143-160. doi: 10.1007/978-1-4614-1260-1_7.
[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.
[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.
[16]	A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 1999, 23-37.
[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.
[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.
[19]	D. G. Ebin, On the space of Riemannian metrics, Bull. Amer. Math. Soc., 74 (1968), 1001-1003. doi: 10.1090/S0002-9904-1968-12115-9.
[20]	D. G. Ebin, The manifold of Riemannian metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970, 11-40.
[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.
[22]	H. I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194. doi: 10.4310/jdg/1214427887.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009.
[30]	F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braş sov Ser. Ⅲ, 2 (2009), 55-58.
[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.
[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.
[33]	P. Iglesias-Zemmour, Diffeology, vol. 185 of Mathematical Surveys and Monographs, American Mathematical Society, 2013. doi: 10.1090/surv/185.
[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.
[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.
[36]	H. H. Keller, Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics, Vol. 417, Springer-Verlag, Berlin-New York, 1974.
[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.
[38]	B. Khesin and V. Ovsienko, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.
[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.
[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.
[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.
[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.
[43]	J. Lenells, The Hunter-Saxton equation: A geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. doi: 10.1137/050647451.
[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.
[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.
[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.
[47]	P. W. Michor, Manifolds of Differentiable Mappings, vol. 3 of Shiva Mathematics Series, Shiva Publishing Ltd., Nantwich, 1980.
[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.
[49]	P. Michor and D. Mumford, On Euler's equation and 'EPDiff', The Journal of Geometric Mechanics, 5 (2013), 319-344. doi: 10.3934/jgm.2013.5.319.
[50]	J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984,1007-1057.
[51]	G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[52]	G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104. doi: 10.1007/PL00012648.
[53]	G. Misiolek and S. C. Preston, Fredholm properties of Riemannian exponential maps on diffeomorphism groups, Invent. Math., 179 (2010), 191-227. doi: 10.1007/s00222-009-0217-3.
[54]	J. J. Moreau, Une méthode de "cinématique fonctionnelle" en hydrodynamique, C. R. Acad. Sci. Paris, 249 (1959), 2156-2158.
[55]	H. Omori, On the group of diffeomorphisms on a compact manifold, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970,167-183.
[56]	H. Omori, Infinite-dimensional Lie Groups, vol. 158 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author.
[57]	R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[58]	H. Poincaré, Sur une nouvelle forme des équations de la mécanique, C.R. Acad. Sci., 132 (1901), 369-371.
[59]	M. Ruzhansky and V. Turunen, Pseudo-differential Operators and Symmetries, vol. 2 of Pseudo-Differential Operators. Theory and Applications, Birkhäuser Verlag, Basel, 2010, Background analysis and advanced topics. doi: 10.1007/978-3-7643-8514-9.
[60]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335.
[61]	S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom., 55 (2000), 145-191. doi: 10.4310/jdg/1090340568.
[62]	A. Shnirelman, Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, 4 (1994), 586-620. doi: 10.1007/BF01896409.
[63]	A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128 (1985), 82-109,144.
[64]	A. Trouvé and L. Younes, Local geometry of deformable templates, SIAM J. Math. Anal., 37 (2005), 17-59 (electronic). doi: 10.1137/S0036141002404838.
[65]	M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric, J. Nonlinear Math. Phys., 17 (2010), 7-11. doi: 10.1142/S1402925110000544.

[1]	Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335
[2]	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
[3]	Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[4]	Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389
[5]	Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365
[6]	Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323
[7]	Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125
[8]	Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[9]	Rafael de la Llave, A. Windsor. Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1141-1154. doi: 10.3934/dcds.2011.29.1141
[10]	Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. The groups of order at most 2000. Electronic Research Announcements, 2001, 7: 1-4.
[11]	Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations and Control Theory, 2020, 9 (3) : 773-793. doi: 10.3934/eect.2020033
[12]	Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241
[13]	Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
[14]	Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[15]	Ruy Coimbra Charão, Juan Torres Espinoza, Ryo Ikehata. A second order fractional differential equation under effects of a super damping. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4433-4454. doi: 10.3934/cpaa.2020202
[16]	Ru-Yu Lai, Laurel Ohm. Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations. Inverse Problems and Imaging, 2022, 16 (2) : 305-323. doi: 10.3934/ipi.2021051
[17]	Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure and Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587
[18]	Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120
[19]	Sang-Gyun Youn. On the Sobolev embedding properties for compact matrix quantum groups of Kac type. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3341-3366. doi: 10.3934/cpaa.2020148
[20]	Flavia Antonacci, Marco Degiovanni. On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 833-842. doi: 10.3934/dcds.2006.15.833

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Boris Kolev
on Google Scholar
Boris Kolev

2021 Impact Factor: 0.737

Tools

Article outline

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Boris Kolev
on Google Scholar
Boris Kolev

[Back to Top]