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.

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. BauerJ. 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. BauerB. 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

[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.  Google Scholar

[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.  Google Scholar

[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]

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

[11]

A. ConstantinT. KappelerB. 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]

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

[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.  Google Scholar

[14]

P. ConstantinP. 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. DegasperisD. 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]

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.  Google Scholar

[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.  Google Scholar

[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. EscherB. 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. LenellsG. 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]

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.  Google Scholar

[50]

J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984,1007-1057.  Google Scholar

[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.  Google Scholar

[52]

G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104.  doi: 10.1007/PL00012648.  Google Scholar

[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.  Google Scholar

[54]

J. J. Moreau, Une méthode de "cinématique fonctionnelle" en hydrodynamique, C. R. Acad. Sci. Paris, 249 (1959), 2156-2158.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[57]

R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.  Google Scholar

[58]

H. Poincaré, Sur une nouvelle forme des équations de la mécanique, C.R. Acad. Sci., 132 (1901), 369-371.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[62]

A. Shnirelman, Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, 4 (1994), 586-620.  doi: 10.1007/BF01896409.  Google Scholar

[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.   Google Scholar

[64]

A. Trouvé and L. Younes, Local geometry of deformable templates, SIAM J. Math. Anal., 37 (2005), 17-59 (electronic).  doi: 10.1137/S0036141002404838.  Google Scholar

[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.  Google Scholar

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. BauerJ. 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. BauerB. 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

[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.  Google Scholar

[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.  Google Scholar

[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]

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

[11]

A. ConstantinT. KappelerB. 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]

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

[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.  Google Scholar

[14]

P. ConstantinP. 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. DegasperisD. 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]

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.  Google Scholar

[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.  Google Scholar

[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. EscherB. 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. LenellsG. 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]

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.  Google Scholar

[50]

J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984,1007-1057.  Google Scholar

[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.  Google Scholar

[52]

G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104.  doi: 10.1007/PL00012648.  Google Scholar

[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.  Google Scholar

[54]

J. J. Moreau, Une méthode de "cinématique fonctionnelle" en hydrodynamique, C. R. Acad. Sci. Paris, 249 (1959), 2156-2158.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[57]

R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.  Google Scholar

[58]

H. Poincaré, Sur une nouvelle forme des équations de la mécanique, C.R. Acad. Sci., 132 (1901), 369-371.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[62]

A. Shnirelman, Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, 4 (1994), 586-620.  doi: 10.1007/BF01896409.  Google Scholar

[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.   Google Scholar

[64]

A. Trouvé and L. Younes, Local geometry of deformable templates, SIAM J. Math. Anal., 37 (2005), 17-59 (electronic).  doi: 10.1137/S0036141002404838.  Google Scholar

[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.  Google Scholar

[1]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[2]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[3]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[4]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[5]

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

[6]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[7]

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

[8]

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

[9]

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

[10]

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

[11]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017

[12]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[13]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[14]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[15]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[16]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[17]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278

[18]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[19]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[20]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (34)
  • HTML views (74)
  • Cited by (7)

Other articles
by authors

[Back to Top]