# American Institute of Mathematical Sciences

November  2011, 30(4): 1161-1180. doi: 10.3934/dcds.2011.30.1161

## Some remarks for a modified periodic Camassa-Holm system

 1 Department of Mathematics, Southeast University, Nanjing 210018 2 Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

Received  May 2010 Revised  August 2010 Published  May 2011

This paper is concerned with a modified two-component periodic Camassa-Holm system. The local well-posedness and low regularity result of solution are established by using the techniques of pseudoparabolic regularization and some priori estimates derived from the equation itself. A wave-breaking for strong solutions and several results of blow-up solution with certain initial profiles are described. In addition, the initial boundary value problem for a modified two-component periodic Camassa-Holm system is also considered.
Citation: Guangying Lv, Mingxin Wang. Some remarks for a modified periodic Camassa-Holm system. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1161-1180. doi: 10.3934/dcds.2011.30.1161
##### References:
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Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar [13] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar [14] A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1245. doi: 10.1512/iumj.1998.47.1466.  Google Scholar [15] A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar [16] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar [17] A. Constantin, V. Gerdjikov and R. I. 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Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [22] 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.  Google Scholar [23] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1998), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar [24] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.  Google Scholar [25] G. Coclite, H. Holden and K. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.  Google Scholar [26] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.  Google Scholar [27] J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.  Google Scholar [28] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equation, 33 (2008), 377-395.  Google Scholar [29] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.  Google Scholar [30] Y. Fu and C. Qu, Well-posedness and blow-up solution for a new coupled Camassa-Holm system with peakons, J. Math. Phys., 50 (2009).  Google Scholar [31] Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448. doi: 10.1007/s00208-010-0483-9.  Google Scholar [32] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014.  Google Scholar [33] C. Guan, K. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemporary Mathematics, 526 (2010). Google Scholar [34] G. Gui and Y. Liu, On the cauchy problem for the two-component Camassa-Holm system,, Math. Z., ().  doi: 10.1007/s00209-009-0660-2.  Google Scholar [35] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.  Google Scholar [36] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597.  Google Scholar [37] H. Holden and X. Raynaud, Global conservation solutions of the Camassa-Holm equation-A lagrangian point of view, Commun. Partial Differential Equation, 32 (2007), 1511-1549.  Google Scholar [38] D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two component Camassa-Holm equation, Phy. Rev. E, 79 (2009), 1-13. doi: 10.1103/PhysRevE.79.016601.  Google Scholar [39] D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1.  Google Scholar [40] R. Iorio and de Magãlhaes Iorio, "Fourier Analysis and Partial Differential Equation," Cambridge University Press, Cambridge, 2001.  Google Scholar [41] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82. Google Scholar [42] T. Kato, Quasi-linear equation of evolution, with application to partial differential equation, in "Spectral Theory and Differential Equations," Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.  Google Scholar [43] 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.  Google Scholar [44] M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves" (Ed. A. Kundu), Springer, Berlin, (2007), 31-49. doi: 10.1007/978-3-540-71256-5_2.  Google Scholar [45] Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  Google Scholar [46] J. L. Lions, "Quelques Méthodes de Résolution des Problèaux Limites Nonlinéaires," Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [47] H. P. McKean, Integrable systems and algebraic curves, Lecture Notes in Math., 755, Springer, Berlin, (1979), 83-200.  Google Scholar [48] 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.  Google Scholar [49] V. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen, 21 (1987), 81-82.  Google Scholar [50] H. Segur, Waves in shallow water, with emphasis int the tsunami of 2004, in "Tsunami and Nonlinear Waves" (ed. A. Kundu), Springer, Berlin, (2007), 31-49.  Google Scholar [51] W. Walter, "Differential and Integral Inequalities," Springer, New York, 1970.  Google Scholar [52] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar [53] Z. Yin, Well-posedness, global existence phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 10 (2004), 393-411. doi: 10.3934/dcds.2004.11.393.  Google Scholar [54] Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6.  Google Scholar

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##### References:
 [1] J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London A, 278 (1975), 555-601. doi: 10.1098/rsta.1975.0035.  Google Scholar [2] A. Bressan and A. Constantin, Global conservation solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar [4] R. Cammasa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar [5] M. Chen, S. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15. doi: 10.1007/s11005-005-0041-7.  Google Scholar [6] A. Constantin, The hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.  Google Scholar [7] A. Constantin, On the inverse spectral problem for the Camasa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.  Google Scholar [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier., 50 (2000), 321-362.  Google Scholar [9] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar [10] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar [11] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 1992 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [12] A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar [13] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar [14] A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1245. doi: 10.1512/iumj.1998.47.1466.  Google Scholar [15] A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar [16] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar [17] A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.  Google Scholar [18] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A., 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.  Google Scholar [19] A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity over variable depth, with applicastions to tsuami, Fluid Dynam. Res., 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar [20] 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.  Google Scholar [21] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equation, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [22] 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.  Google Scholar [23] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1998), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar [24] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.  Google Scholar [25] G. Coclite, H. Holden and K. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.  Google Scholar [26] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.  Google Scholar [27] J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.  Google Scholar [28] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equation, 33 (2008), 377-395.  Google Scholar [29] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.  Google Scholar [30] Y. Fu and C. Qu, Well-posedness and blow-up solution for a new coupled Camassa-Holm system with peakons, J. Math. Phys., 50 (2009).  Google Scholar [31] Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448. doi: 10.1007/s00208-010-0483-9.  Google Scholar [32] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014.  Google Scholar [33] C. Guan, K. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemporary Mathematics, 526 (2010). Google Scholar [34] G. Gui and Y. Liu, On the cauchy problem for the two-component Camassa-Holm system,, Math. Z., ().  doi: 10.1007/s00209-009-0660-2.  Google Scholar [35] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.  Google Scholar [36] D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597.  Google Scholar [37] H. Holden and X. Raynaud, Global conservation solutions of the Camassa-Holm equation-A lagrangian point of view, Commun. Partial Differential Equation, 32 (2007), 1511-1549.  Google Scholar [38] D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two component Camassa-Holm equation, Phy. Rev. E, 79 (2009), 1-13. doi: 10.1103/PhysRevE.79.016601.  Google Scholar [39] D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1.  Google Scholar [40] R. Iorio and de Magãlhaes Iorio, "Fourier Analysis and Partial Differential Equation," Cambridge University Press, Cambridge, 2001.  Google Scholar [41] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82. Google Scholar [42] T. Kato, Quasi-linear equation of evolution, with application to partial differential equation, in "Spectral Theory and Differential Equations," Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.  Google Scholar [43] 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.  Google Scholar [44] M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves" (Ed. A. Kundu), Springer, Berlin, (2007), 31-49. doi: 10.1007/978-3-540-71256-5_2.  Google Scholar [45] Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  Google Scholar [46] J. L. Lions, "Quelques Méthodes de Résolution des Problèaux Limites Nonlinéaires," Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [47] H. P. 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