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Suspension of the billiard maps in the Lazutkin's coordinate
Wave breaking and global existence for the periodic rotation-Camassa-Holm system
School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China |
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.
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. |
show all references
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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