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Euler equations on a semi-direct product of the diffeomorphisms group by itself
1. | Institute for Applied Mathematics, University of Hanover, D-30167 Hanover, Germany |
2. | School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland |
3. | LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13 |
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. |
[2] |
V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics,", Applied Mathematical Sciences, 125 (1998).
|
[3] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl. (Singapore), 5 (2007), 1.
doi: 10.1142/S0219530507000857. |
[4] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.
doi: 10.1007/s00205-006-0010-z. |
[5] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.
doi: 10.5802/aif.1757. |
[7] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[8] |
A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.
doi: 10.1007/PL00004793. |
[9] |
A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.
doi: 10.1016/j.physleta.2008.10.050. |
[10] |
A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002).
doi: 10.1088/0305-4470/35/32/201. |
[11] |
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. |
[12] |
A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle,, J. Nonlinear Sci., 16 (2006), 109.
doi: 10.1007/s00332-005-0707-4. |
[13] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165.
doi: 10.1007/s00205-008-0128-2. |
[14] |
C. Cotter, D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation,, J. Phys. A: Math. Theor., 44 (2011), 265.
doi: 10.1088/1751-8113/44/26/265205. |
[15] |
J. Escher, Non-metric two-component Euler equations on the circle,, Monatshefte für Mathematik, (2011).
doi: 10.1007/s00605-011-0323-3. |
[16] |
I. M. Gel'fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle,, Funkcional. Anal. i Priložen, 2 (1968), 92.
|
[17] |
R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.
|
[18] |
D. J. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342.
doi: 10.2991/jnmp.2005.12.3.3. |
[19] |
D. D. Holm and R. I. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications,, J. of Geometry and Symmetry in Physics, 22 (2011), 13. Google Scholar |
[20] |
D. D. Holm and J. E.Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,, in, 232 (2005), 203.
|
[21] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.
doi: 10.1006/aima.1998.1721. |
[22] |
R. I. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 365 (2007), 2267.
doi: 10.1098/rsta.2007.2007. |
[23] |
R. I. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case,, Wave Motion, 46 (2009), 389.
doi: 10.1016/j.wavemoti.2009.06.012. |
[24] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and Related Models for Water Waves,, J. Fluid Mech., 455 (2002), 63.
doi: 10.1017/S0022112001007224. |
[25] |
B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116.
doi: 10.1016/S0001-8708(02)00063-4. |
[26] |
B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2333.
|
[27] |
G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.
|
[28] |
I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,", Progress in Mathematics, 118 (1994).
|
[29] |
P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Not. IMRN, 2010 (): 1981.
|
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. |
[2] |
V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics,", Applied Mathematical Sciences, 125 (1998).
|
[3] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl. (Singapore), 5 (2007), 1.
doi: 10.1142/S0219530507000857. |
[4] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.
doi: 10.1007/s00205-006-0010-z. |
[5] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.
doi: 10.5802/aif.1757. |
[7] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[8] |
A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.
doi: 10.1007/PL00004793. |
[9] |
A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.
doi: 10.1016/j.physleta.2008.10.050. |
[10] |
A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002).
doi: 10.1088/0305-4470/35/32/201. |
[11] |
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. |
[12] |
A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle,, J. Nonlinear Sci., 16 (2006), 109.
doi: 10.1007/s00332-005-0707-4. |
[13] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165.
doi: 10.1007/s00205-008-0128-2. |
[14] |
C. Cotter, D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation,, J. Phys. A: Math. Theor., 44 (2011), 265.
doi: 10.1088/1751-8113/44/26/265205. |
[15] |
J. Escher, Non-metric two-component Euler equations on the circle,, Monatshefte für Mathematik, (2011).
doi: 10.1007/s00605-011-0323-3. |
[16] |
I. M. Gel'fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle,, Funkcional. Anal. i Priložen, 2 (1968), 92.
|
[17] |
R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.
|
[18] |
D. J. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342.
doi: 10.2991/jnmp.2005.12.3.3. |
[19] |
D. D. Holm and R. I. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications,, J. of Geometry and Symmetry in Physics, 22 (2011), 13. Google Scholar |
[20] |
D. D. Holm and J. E.Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,, in, 232 (2005), 203.
|
[21] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.
doi: 10.1006/aima.1998.1721. |
[22] |
R. I. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 365 (2007), 2267.
doi: 10.1098/rsta.2007.2007. |
[23] |
R. I. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case,, Wave Motion, 46 (2009), 389.
doi: 10.1016/j.wavemoti.2009.06.012. |
[24] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and Related Models for Water Waves,, J. Fluid Mech., 455 (2002), 63.
doi: 10.1017/S0022112001007224. |
[25] |
B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116.
doi: 10.1016/S0001-8708(02)00063-4. |
[26] |
B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2333.
|
[27] |
G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.
|
[28] |
I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,", Progress in Mathematics, 118 (1994).
|
[29] |
P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Not. IMRN, 2010 (): 1981.
|
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