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Local well-posedness of the EPDiff equation: A survey
Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France |
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.
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. 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. |
[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. |
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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. |
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A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724.
doi: 10.1002/cpa.3160380605. |
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A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 170-183.
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Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.(2), 92 (1970), 102-163.
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A concise presentation of the Euler equations of hydrodynamics, Comm. Partial Differential Equations, 9 (1984), 539-559.
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H. I. Elĭasson,
Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194.
doi: 10.4310/jdg/1214427887. |
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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. |
[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. 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. |
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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.
|
[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. |
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H. H. Keller,
Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics, Vol. 417, Springer-Verlag, Berlin-New York, 1974. |
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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. |
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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. |
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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. |
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S. Lang,
Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1999.
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The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064.
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J. Lenells,
The Hunter-Saxton equation: A geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277.
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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. |
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A. J. Majda and A. L. Bertozzi,
Vorticity and Incompressible Flow, vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. |
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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. |
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P. W. Michor,
Manifolds of Differentiable Mappings, vol. 3 of Shiva Mathematics Series, Shiva Publishing Ltd., Nantwich, 1980. |
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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. |
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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. |
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J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984,1007-1057. |
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A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
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G. Misiolek,
Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104.
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G. Misiolek and S. C. Preston,
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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. |
[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. 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. |
[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. 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. |
[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.
|
[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,
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