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 and Continuous Dynamical Systems, 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), 319-361. doi: 10.5802/aif.233.

[2]

A. Bastiani, Applications différentiable et variétés différentiables de dimension infinie, J. Anal. Math., 13 (1964), 1-114. doi: 10.1007/BF02786619.

[3]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vis., 50 (2014), 60-97. doi: 10.1007/s10851-013-0490-z.

[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), 2010-2053. doi: 10.1016/j.jde.2014.11.021.

[5]

E. C. Cismas, Euler-Poincaré equations on semi-direct products, Monatshefte für Math., 179 (2014), 491-507. doi: 10.1007/s00605-014-0720-5.

[6]

E. C. Cismas, https://www.tib.eu/en/search/download/?tx_tibsearch_search Ph.D thesis, Leibniz University in Hannover, 2015.

[7]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), 51-79. doi: 10.1088/0305-4470/35/32/201.

[8]

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.

[9]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 2 (1970), 102-163. doi: 10.2307/1970699.

[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) 335-372. doi: 10.3934/jgm.2014.6.335.

[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), 313-322.

[12]

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.

[13]

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.

[14]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, Communications in Contemporary Mathematics, 14 (2012), 24-36. doi: 10.1142/S0219199712500162.

[15]

J. Escher and B. Kolev, Geometrical methods for equations of hydrodynamical type, J. Nonlinear Math. Phys., 19 (2012), 161-178. doi: 10.1142/S140292511240013X.

[16]

L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro, Les Publications CRM, Montreal, QC, 2007.

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

[18]

A. Hirani, J. Marsden and J. Arvo, Averaged template matching equations, Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science, 2134 (2001), 528-543. doi: 10.1007/3-540-44745-8_35.

[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), 1-81. doi: 10.1006/aima.1998.1721.

[20]

D. Holm and J. Marsden, Momentum maps and measure-valued solutions for the EPDiff equation, The Breadth of Symplectic and Poisson geometry, A festschrift for Alan Weinstein, Progress in Mathematics, 232 (2004), 203-235.

[21]

H. H. Keller, Differential Calculus in Locally Convex Spaces, Lecture Notes in Math., Springer-Verlag, 1974. doi: 10.1007/BFb0070564.

[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), 101-123. doi: 10.1007/BFb0066026.

[23]

M. Kohlmann, On a two-component Camassa-Holm system, Journal of Geometry and Physics, 62 (2012), 832-838. doi: 10.1016/j.geomphys.2012.01.001.

[24]

B. Kolev, Lie groups and mechanics: An introduction, J. Nonlinear Math. Phys., 11 (2004), 480-498.

[25]

S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, 191, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.

[26]

A. D. Michal, Differentiable calculus in linear topological spaces, Proc. Natl. Acad. Sci., 24 (1938), 340-342.

[27]

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

[28]

J. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, II (Les Houches), North-Holland, Amsterdam, (1984), 1007-1057.

[29]

G. Misiołek 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.

[30]

O. Muller, A metric approach to Fréchet geometry, J. Geom. Phys., 58 (2008), 1477-1500. doi: 10.1016/j.geomphys.2008.06.004.

[31]

K. H. Neeb, Towards a Lie theory of locally convex groups. Japan. J. Math., 1 (2006), 291-468. doi: 10.1007/s11537-006-0606-y.

[32]

H. Omori, Infinite-dimensional Lie Groups, Translations of Math. Monographs, 158, 1997.

[33]

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

[34]

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

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]

A. Bastiani, Applications différentiable et variétés différentiables de dimension infinie, J. Anal. Math., 13 (1964), 1-114. doi: 10.1007/BF02786619.

[3]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vis., 50 (2014), 60-97. doi: 10.1007/s10851-013-0490-z.

[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), 2010-2053. doi: 10.1016/j.jde.2014.11.021.

[5]

E. C. Cismas, Euler-Poincaré equations on semi-direct products, Monatshefte für Math., 179 (2014), 491-507. doi: 10.1007/s00605-014-0720-5.

[6]

E. C. Cismas, https://www.tib.eu/en/search/download/?tx_tibsearch_search Ph.D thesis, Leibniz University in Hannover, 2015.

[7]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), 51-79. doi: 10.1088/0305-4470/35/32/201.

[8]

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.

[9]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 2 (1970), 102-163. doi: 10.2307/1970699.

[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) 335-372. doi: 10.3934/jgm.2014.6.335.

[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), 313-322.

[12]

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.

[13]

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.

[14]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, Communications in Contemporary Mathematics, 14 (2012), 24-36. doi: 10.1142/S0219199712500162.

[15]

J. Escher and B. Kolev, Geometrical methods for equations of hydrodynamical type, J. Nonlinear Math. Phys., 19 (2012), 161-178. doi: 10.1142/S140292511240013X.

[16]

L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro, Les Publications CRM, Montreal, QC, 2007.

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

[18]

A. Hirani, J. Marsden and J. Arvo, Averaged template matching equations, Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science, 2134 (2001), 528-543. doi: 10.1007/3-540-44745-8_35.

[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), 1-81. doi: 10.1006/aima.1998.1721.

[20]

D. Holm and J. Marsden, Momentum maps and measure-valued solutions for the EPDiff equation, The Breadth of Symplectic and Poisson geometry, A festschrift for Alan Weinstein, Progress in Mathematics, 232 (2004), 203-235.

[21]

H. H. Keller, Differential Calculus in Locally Convex Spaces, Lecture Notes in Math., Springer-Verlag, 1974. doi: 10.1007/BFb0070564.

[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), 101-123. doi: 10.1007/BFb0066026.

[23]

M. Kohlmann, On a two-component Camassa-Holm system, Journal of Geometry and Physics, 62 (2012), 832-838. doi: 10.1016/j.geomphys.2012.01.001.

[24]

B. Kolev, Lie groups and mechanics: An introduction, J. Nonlinear Math. Phys., 11 (2004), 480-498.

[25]

S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, 191, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.

[26]

A. D. Michal, Differentiable calculus in linear topological spaces, Proc. Natl. Acad. Sci., 24 (1938), 340-342.

[27]

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

[28]

J. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, II (Les Houches), North-Holland, Amsterdam, (1984), 1007-1057.

[29]

G. Misiołek 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.

[30]

O. Muller, A metric approach to Fréchet geometry, J. Geom. Phys., 58 (2008), 1477-1500. doi: 10.1016/j.geomphys.2008.06.004.

[31]

K. H. Neeb, Towards a Lie theory of locally convex groups. Japan. J. Math., 1 (2006), 291-468. doi: 10.1007/s11537-006-0606-y.

[32]

H. Omori, Infinite-dimensional Lie Groups, Translations of Math. Monographs, 158, 1997.

[33]

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

[34]

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

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