# American Institute of Mathematical Sciences

September  2015, 7(3): 281-293. doi: 10.3934/jgm.2015.7.281

## Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach

 1 Institute for Applied Mathematics, University of Hanover, D-30167 Hanover 2 School of Mathematical Sciences, University College Cork, Cork, Ireland

Received  March 2015 Revised  June 2015 Published  July 2015

In the following we study the qualitative properties of solutions to the geodesic flow induced by a higher order two-component Camassa-Holm system. In particular, criteria to ensure the existence of temporally global solutions are presented. Moreover in the metric case, and for inertia operators of order higher than three, the flow is shown to be geodesically complete.
Citation: Joachim Escher, Tony Lyons. Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach. Journal of Geometric Mechanics, 2015, 7 (3) : 281-293. doi: 10.3934/jgm.2015.7.281
##### 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] 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. Google Scholar [3] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar [4] 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. Google Scholar [5] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, Volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), (2011). doi: 10.1137/1.9781611971873. Google Scholar [6] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [7] A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar [8] A. Constantin and R. I. 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. Google Scholar [9] 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. Google Scholar [10] 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 [11] A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle,, J. Nonlin. Math. Phys., 10 (2003), 424. doi: 10.2991/jnmp.2003.10.4.1. Google Scholar [12] A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and perturbation theory (Rome, (1999), 23. Google Scholar [13] D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math., 92 (1970), 102. doi: 10.2307/1970699. Google Scholar [14] J. Escher, Non-metric two-component Euler equations on the circle,, Monatsh. Math., 167 (2012), 449. doi: 10.1007/s00605-011-0323-3. Google Scholar [15] J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity,, Ann. Mat. Pur. App., (2014). doi: 10.1007/s10231-014-0461-z. Google Scholar [16] 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 [17] J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle,, J. Evol. Equ., 6 (2014), 335. doi: 10.1007/s00028-014-0245-3. Google Scholar [18] J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle,, J. Geom. Mech., 14 (2014), 949. doi: 10.3934/jgm.2014.6.335. Google Scholar [19] J. Escer and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the $b$-equation,, J. Reine Angew. Math., 624 (2008), 51. doi: 10.1515/CRELLE.2008.080. Google Scholar [20] D. Henry, Compactly supported solutions of a family of nonlinear differential equations,, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 15 (2008), 145. Google Scholar [21] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Dyn. Syst. Ser. B, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar [22] D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlin. Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar [23] R. 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. Google Scholar [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. Google Scholar [25] R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dynam. Res., 33 (2003), 97. doi: 10.1016/S0169-5983(03)00036-4. Google Scholar [26] B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation,, Wave Motion, 46 (2009), 412. doi: 10.1016/j.wavemoti.2009.06.005. Google Scholar [27] G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar

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##### 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] 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. Google Scholar [3] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar [4] 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. Google Scholar [5] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, Volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), (2011). doi: 10.1137/1.9781611971873. Google Scholar [6] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [7] A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar [8] A. Constantin and R. I. 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. Google Scholar [9] 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. Google Scholar [10] 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 [11] A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle,, J. Nonlin. Math. Phys., 10 (2003), 424. doi: 10.2991/jnmp.2003.10.4.1. Google Scholar [12] A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and perturbation theory (Rome, (1999), 23. Google Scholar [13] D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math., 92 (1970), 102. doi: 10.2307/1970699. Google Scholar [14] J. Escher, Non-metric two-component Euler equations on the circle,, Monatsh. Math., 167 (2012), 449. doi: 10.1007/s00605-011-0323-3. Google Scholar [15] J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity,, Ann. Mat. Pur. App., (2014). doi: 10.1007/s10231-014-0461-z. Google Scholar [16] 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 [17] J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle,, J. Evol. Equ., 6 (2014), 335. doi: 10.1007/s00028-014-0245-3. Google Scholar [18] J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle,, J. Geom. Mech., 14 (2014), 949. doi: 10.3934/jgm.2014.6.335. Google Scholar [19] J. Escer and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the $b$-equation,, J. Reine Angew. Math., 624 (2008), 51. doi: 10.1515/CRELLE.2008.080. Google Scholar [20] D. Henry, Compactly supported solutions of a family of nonlinear differential equations,, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 15 (2008), 145. Google Scholar [21] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Dyn. Syst. Ser. B, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar [22] D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlin. Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar [23] R. 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. Google Scholar [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. Google Scholar [25] R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dynam. Res., 33 (2003), 97. doi: 10.1016/S0169-5983(03)00036-4. Google Scholar [26] B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation,, Wave Motion, 46 (2009), 412. doi: 10.1016/j.wavemoti.2009.06.005. Google Scholar [27] G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar
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