November  2016, 36(11): 5993-6022. doi: 10.3934/dcds.2016063

Euler-Poincaré-Arnold equations on semi-direct products II

1. 

Institute for Applied Mathematics, Leibniz University, Hannover, Welfengarten 1, 30167, Germany

Received  December 2015 Revised  June 2016 Published  August 2016

We study the well-posedness of the Euler-Poincaré-Arnold equations on the semi-direct products of the group of orientation-preserving diffeomorphisms of the circle with itself. To achieve this goal, according to the previous results obtained in [5], we had to extend the results obtained in [10] for the general case of inertia operators of pseudo-differential type.
Citation: Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063
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.  doi: 10.5802/aif.233.  Google Scholar

[2]

A. Bastiani, Applications différentiable et variétés différentiables de dimension infinie,, J. Anal. Math., 13 (1964), 1.  doi: 10.1007/BF02786619.  Google Scholar

[3]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups,, J. Math. Imaging Vis., 50 (2014), 60.  doi: 10.1007/s10851-013-0490-z.  Google Scholar

[4]

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

[5]

E. C. Cismas, Euler-Poincaré equations on semi-direct products,, Monatshefte für Math., 179 (2014), 491.  doi: 10.1007/s00605-014-0720-5.  Google Scholar

[6]

E. C. Cismas, https://www.tib.eu/en/search/download/?tx_tibsearch_search, Ph.D thesis, (2015).   Google Scholar

[7]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, \emph{J. Phys. A}, 35 (2002), 51.  doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

J. Escher, R. Ivanov and B. Kolev, Euler equations on a semi-direct product of the diffeomorphims group by itself,, Journal of Geometric Mechanics, 3 (2011), 313.   Google Scholar

[12]

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

[13]

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

[14]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012), 24.  doi: 10.1142/S0219199712500162.  Google Scholar

[15]

J. Escher and B. Kolev, Geometrical methods for equations of hydrodynamical type,, J. Nonlinear Math. Phys., 19 (2012), 161.  doi: 10.1142/S140292511240013X.  Google Scholar

[16]

L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro,, Les Publications CRM, (2007).   Google Scholar

[17]

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

[18]

A. Hirani, J. Marsden and J. Arvo, Averaged template matching equations,, Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition, 2134 (2001), 528.  doi: 10.1007/3-540-44745-8_35.  Google Scholar

[19]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[20]

D. Holm and J. Marsden, Momentum maps and measure-valued solutions for the EPDiff equation,, The Breadth of Symplectic and Poisson geometry, 232 (2004), 203.   Google Scholar

[21]

H. H. Keller, Differential Calculus in Locally Convex Spaces,, Lecture Notes in Math., (1974).  doi: 10.1007/BFb0070564.  Google Scholar

[22]

A. A. Kirillov, Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments,, Twistor Geometry and Non-Linear Systems, 970 (1982), 101.  doi: 10.1007/BFb0066026.  Google Scholar

[23]

M. Kohlmann, On a two-component Camassa-Holm system,, Journal of Geometry and Physics, 62 (2012), 832.  doi: 10.1016/j.geomphys.2012.01.001.  Google Scholar

[24]

B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480.   Google Scholar

[25]

S. Lang, Fundamentals of Differential Geometry,, Graduate Texts in Mathematics, 191 (1999).  doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[26]

A. D. Michal, Differentiable calculus in linear topological spaces,, Proc. Natl. Acad. Sci., 24 (1938), 340.   Google Scholar

[27]

P. Michor and A. Kriegl, The Convenient Setting of Global Analysis,, Math. Surveys and Monographs, 53 (1997).  doi: 10.1090/surv/053.  Google Scholar

[28]

J. Milnor, Remarks on infinite-dimensional Lie groups,, Relativity, (1984), 1007.   Google Scholar

[29]

G. Misiołek and S. C. Preston, Fredholm properties of Riemannian exponential maps on diffeomorphism groups,, Invent. math., 179 (2010), 191.  doi: 10.1007/s00222-009-0217-3.  Google Scholar

[30]

O. Muller, A metric approach to Fréchet geometry,, J. Geom. Phys., 58 (2008), 1477.  doi: 10.1016/j.geomphys.2008.06.004.  Google Scholar

[31]

K. H. Neeb, Towards a Lie theory of locally convex groups., Japan. J. Math., 1 (2006), 291.  doi: 10.1007/s11537-006-0606-y.  Google Scholar

[32]

H. Omori, Infinite-dimensional Lie Groups,, Translations of Math. Monographs, 158 (1997).   Google Scholar

[33]

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

[34]

M. Ruzhansky and V. Turunen, Pseudo-differential Operators and Symmetries,, Birkhauser, (2010).  doi: 10.1007/978-3-7643-8514-9.  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.  doi: 10.5802/aif.233.  Google Scholar

[2]

A. Bastiani, Applications différentiable et variétés différentiables de dimension infinie,, J. Anal. Math., 13 (1964), 1.  doi: 10.1007/BF02786619.  Google Scholar

[3]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups,, J. Math. Imaging Vis., 50 (2014), 60.  doi: 10.1007/s10851-013-0490-z.  Google Scholar

[4]

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

[5]

E. C. Cismas, Euler-Poincaré equations on semi-direct products,, Monatshefte für Math., 179 (2014), 491.  doi: 10.1007/s00605-014-0720-5.  Google Scholar

[6]

E. C. Cismas, https://www.tib.eu/en/search/download/?tx_tibsearch_search, Ph.D thesis, (2015).   Google Scholar

[7]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, \emph{J. Phys. A}, 35 (2002), 51.  doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

J. Escher, R. Ivanov and B. Kolev, Euler equations on a semi-direct product of the diffeomorphims group by itself,, Journal of Geometric Mechanics, 3 (2011), 313.   Google Scholar

[12]

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

[13]

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

[14]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012), 24.  doi: 10.1142/S0219199712500162.  Google Scholar

[15]

J. Escher and B. Kolev, Geometrical methods for equations of hydrodynamical type,, J. Nonlinear Math. Phys., 19 (2012), 161.  doi: 10.1142/S140292511240013X.  Google Scholar

[16]

L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro,, Les Publications CRM, (2007).   Google Scholar

[17]

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

[18]

A. Hirani, J. Marsden and J. Arvo, Averaged template matching equations,, Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition, 2134 (2001), 528.  doi: 10.1007/3-540-44745-8_35.  Google Scholar

[19]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[20]

D. Holm and J. Marsden, Momentum maps and measure-valued solutions for the EPDiff equation,, The Breadth of Symplectic and Poisson geometry, 232 (2004), 203.   Google Scholar

[21]

H. H. Keller, Differential Calculus in Locally Convex Spaces,, Lecture Notes in Math., (1974).  doi: 10.1007/BFb0070564.  Google Scholar

[22]

A. A. Kirillov, Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments,, Twistor Geometry and Non-Linear Systems, 970 (1982), 101.  doi: 10.1007/BFb0066026.  Google Scholar

[23]

M. Kohlmann, On a two-component Camassa-Holm system,, Journal of Geometry and Physics, 62 (2012), 832.  doi: 10.1016/j.geomphys.2012.01.001.  Google Scholar

[24]

B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480.   Google Scholar

[25]

S. Lang, Fundamentals of Differential Geometry,, Graduate Texts in Mathematics, 191 (1999).  doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[26]

A. D. Michal, Differentiable calculus in linear topological spaces,, Proc. Natl. Acad. Sci., 24 (1938), 340.   Google Scholar

[27]

P. Michor and A. Kriegl, The Convenient Setting of Global Analysis,, Math. Surveys and Monographs, 53 (1997).  doi: 10.1090/surv/053.  Google Scholar

[28]

J. Milnor, Remarks on infinite-dimensional Lie groups,, Relativity, (1984), 1007.   Google Scholar

[29]

G. Misiołek and S. C. Preston, Fredholm properties of Riemannian exponential maps on diffeomorphism groups,, Invent. math., 179 (2010), 191.  doi: 10.1007/s00222-009-0217-3.  Google Scholar

[30]

O. Muller, A metric approach to Fréchet geometry,, J. Geom. Phys., 58 (2008), 1477.  doi: 10.1016/j.geomphys.2008.06.004.  Google Scholar

[31]

K. H. Neeb, Towards a Lie theory of locally convex groups., Japan. J. Math., 1 (2006), 291.  doi: 10.1007/s11537-006-0606-y.  Google Scholar

[32]

H. Omori, Infinite-dimensional Lie Groups,, Translations of Math. Monographs, 158 (1997).   Google Scholar

[33]

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

[34]

M. Ruzhansky and V. Turunen, Pseudo-differential Operators and Symmetries,, Birkhauser, (2010).  doi: 10.1007/978-3-7643-8514-9.  Google Scholar

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