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]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[2]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

[3]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021025

[4]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[5]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[6]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021063

[7]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016

[8]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[9]

Huan Zhang, Jun Zhou. Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021034

[10]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[11]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[12]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019

[13]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[14]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033

[15]

Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025

[16]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[17]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[18]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

[19]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[20]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (41)
  • HTML views (75)
  • Cited by (6)

Other articles
by authors

[Back to Top]