# American Institute of Mathematical Sciences

April  2017, 37(4): 2243-2257. doi: 10.3934/dcds.2017097

## Wave breaking and global existence for the periodic rotation-Camassa-Holm system

 School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China

Received  September 2016 Revised  November 2016 Published  December 2016

Fund Project: This work is supported by the National Natural Science Foundation of China (No. 11561059).

The rotation-two-componentCamassa-Holm system with the effect of the Coriolis force in therotating fluid is a model in the equatorial water waves. In thispaper we consider its periodic Cauchy problem. The precise blow-upscenarios of strong solutions and several conditions on the initialdata that produce blow-up of the induced solutions are described indetail. Finally, a sufficient condition for global solutions isestablished.

Citation: Ying Zhang. Wave breaking and global existence for the periodic rotation-Camassa-Holm system. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2243-2257. doi: 10.3934/dcds.2017097
##### References:
 [1] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal, 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z. [2] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857. [3] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [4] R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Int. Math. Res. Not., 6 (2011), 1381-1416.  doi: 10.1093/imrn/rnq118. [5] C. Chen and S. Wen, Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system, Discrete Continuous Dynamical Systems, 32 (2012), 3459-3484.  doi: 10.3934/dcds.2012.32.3459. [6] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), 247-253.  doi: 10.1029/2012JC007879. [7] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235.  doi: 10.1006/jdeq.1997.3333. [8] A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399.  doi: 10.1007/s003329910017. [9] A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in shallow water theory, Math. Ann., 312 (1998), 403-416.  doi: 10.1007/s002080050228. [10] A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomenon 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. [11] 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), 75-91.  doi: 10.1007/PL00004793. [12] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [13] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219. [14] A. Constantin and R. Ivanov, On the integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050. [15] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785. [16] R. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 194501.  doi: 10.1103/PhysRevLett.87.194501. [17] J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.  doi: 10.1007/s10231-014-0461-z. [18] J. Escher, O. Lechttenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.  doi: 10.3934/dcds.2007.19.493. [19] L. Fan, H. Gao and Y. Liu, On the rotation-two-component Camassa-Holm system modelling the equatorial water waves, Advances in Mathematics, 291 (2016), 59-89.  doi: 10.1016/j.aim.2015.11.049. [20] F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667.  doi: 10.1007/s00021-014-0175-4. [21] G. L. 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. [22] F. Guo, H. J. Gao and Y. Liu, On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. Lond. Math. Soc., 86 (2012), 810-834.  doi: 10.1112/jlms/jds035. [23] Y. Han, F. Guo and H. J. Gao, On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dullin-Gottwald-Holm system, J. Nonlinear Sci., 23 (2013), 617-656.  doi: 10.1007/s00332-012-9163-0. [24] D. Henry, Equatorially trapped nonlinear water waves in an $β$-plane approximation with centripetal forces J. Fluid Mech. , 804(2016), R1, 11pp. doi: 10.1017/jfm.2016.544. [25] D. Henry and R. Ivanov, One-dimensional weakly nonlinear model equations for the Rossby waves, Discrete Contin. Dyn. Syst. A, 34 (2014), 3025-3034.  doi: 10.3934/dcds.2014.34.3025. [26] H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier, 58 (2008), 945-988.  doi: 10.5802/aif.2375. [27] R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396.  doi: 10.1016/j.wavemoti.2009.06.012. [28] T. Kato, Quasi-linear equations of evolution, with applications to partialdifferential equations spectral theory and differential equation, Lecture Notes in Math., 448 (1975), 25-70. [29] H. P. Mckean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.  doi: 10.4310/AJM.1998.v2.n4.a10. [30] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104.  doi: 10.1007/PL00012648. [31] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900. [32] Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 375-381. [33] P. Z. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not., 2010 (2010), 1981-2021.  doi: 10.1093/imrn/rnp211.

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##### References:
 [1] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal, 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z. [2] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857. [3] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [4] R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Int. Math. Res. Not., 6 (2011), 1381-1416.  doi: 10.1093/imrn/rnq118. [5] C. Chen and S. Wen, Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system, Discrete Continuous Dynamical Systems, 32 (2012), 3459-3484.  doi: 10.3934/dcds.2012.32.3459. [6] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), 247-253.  doi: 10.1029/2012JC007879. [7] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235.  doi: 10.1006/jdeq.1997.3333. [8] A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399.  doi: 10.1007/s003329910017. [9] A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in shallow water theory, Math. Ann., 312 (1998), 403-416.  doi: 10.1007/s002080050228. [10] A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomenon 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. [11] 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), 75-91.  doi: 10.1007/PL00004793. [12] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [13] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219. [14] A. Constantin and R. Ivanov, On the integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050. [15] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785. [16] R. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 194501.  doi: 10.1103/PhysRevLett.87.194501. [17] J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.  doi: 10.1007/s10231-014-0461-z. [18] J. Escher, O. Lechttenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.  doi: 10.3934/dcds.2007.19.493. [19] L. Fan, H. Gao and Y. Liu, On the rotation-two-component Camassa-Holm system modelling the equatorial water waves, Advances in Mathematics, 291 (2016), 59-89.  doi: 10.1016/j.aim.2015.11.049. [20] F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667.  doi: 10.1007/s00021-014-0175-4. [21] G. L. 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. [22] F. Guo, H. J. Gao and Y. Liu, On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, J. Lond. Math. Soc., 86 (2012), 810-834.  doi: 10.1112/jlms/jds035. [23] Y. Han, F. Guo and H. J. Gao, On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dullin-Gottwald-Holm system, J. Nonlinear Sci., 23 (2013), 617-656.  doi: 10.1007/s00332-012-9163-0. [24] D. Henry, Equatorially trapped nonlinear water waves in an $β$-plane approximation with centripetal forces J. Fluid Mech. , 804(2016), R1, 11pp. doi: 10.1017/jfm.2016.544. [25] D. Henry and R. Ivanov, One-dimensional weakly nonlinear model equations for the Rossby waves, Discrete Contin. Dyn. Syst. A, 34 (2014), 3025-3034.  doi: 10.3934/dcds.2014.34.3025. [26] H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier, 58 (2008), 945-988.  doi: 10.5802/aif.2375. [27] R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396.  doi: 10.1016/j.wavemoti.2009.06.012. [28] T. Kato, Quasi-linear equations of evolution, with applications to partialdifferential equations spectral theory and differential equation, Lecture Notes in Math., 448 (1975), 25-70. [29] H. P. Mckean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.  doi: 10.4310/AJM.1998.v2.n4.a10. [30] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104.  doi: 10.1007/PL00012648. [31] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900. [32] Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 375-381. [33] P. Z. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not., 2010 (2010), 1981-2021.  doi: 10.1093/imrn/rnp211.
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