American Institute of Mathematical Sciences

October  2012, 32(10): 3459-3484. doi: 10.3934/dcds.2012.32.3459

Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system

 1 Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, United States 2 Department of Mathematics and Physics, Huaiyin Institute of Technology, Huaiyin, Jiangsu 223003, China

Received  April 2011 Revised  January 2012 Published  May 2012

Considered herein is the generalized two-component periodic Camassa-Holm system. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail. The exact blow-up rates are also determined. Finally, a sufficient condition for global solutions is established.
Citation: Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
References:
 [1] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 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, Analysis and its Applications (Singap.), 5 (2007), 1-27. [3] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, Journal of Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [4] A. Constantin, On the blow-up of solutions of a periodic shallow water equation, Journal of Nonlinear Science, 10 (2000), 391-399. doi: 10.1007/s003329910017. [5] A. Constantin, The trajectories of particles in Stokes waves, Inventiones Mathematicae, 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [6] A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynamics Research, 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004. [7] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [8] A. Constantin and J. Escher, Well-posedness, global existence, and blowup 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. [9] 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. [10] A. Constantin and H. P. Mckean, A shallow water equation on the circle, Communications on Pure and Applied Mathematics, 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [11] A. Constantin and W. A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics, 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. [12] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, Journal of Nonlinear Science, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. [13] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D., 4 (): 47. [14] A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation, Differential Integral Equations, 14 (2001), 821-831. [15] A. Shabat and L. Martínez Alonso, On the prolongation of a hierarchy of hydrodynamic chains, in "New Trends in Integrability and Partial Solvability," NATO Sci. Ser. II Math. Phys. Chem., 132, Kluwer Acad. Publ., Dordrecht, (2004), 263-280. [16] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, Journal of Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002. [17] C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [18] 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. [19] G. B. Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. [20] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, Journal of Functional Analysis, 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. [21] G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Mathematische Zeitschrift, 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2. [22] G. Misiołek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104. doi: 10.1007/PL00012648. [23] G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [24] H. Aratyn, J. F. Gomes and A. H. Zimerman, On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation, SIGMA Symmetry, Integrability and Geometry: Methods and Applications, 2 (2006), Paper 070, 12 pp. [25] H. P. Mckean, Breakdown of a shallow water equation, Asian Journal of Mathematics, 2 (1998), 867-874. [26] 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. [27] J. F. Toland, Stokes waves, Topological Methods in Nonlinear Analysis, 7 (1996), 1-48. [28] J. Lenells, Stability of periodic peakons,, International Mathematics Research Notices, 2004 (): 485. [29] M. Chen, S.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Letters in Mathematical Physics, 75 (2006), 1-15. doi: 10.1007/s11005-005-0041-7. [30] O. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system, Wave Motion, 46 (2009), 397-402. doi: 10.1016/j.wavemoti.2009.06.011. [31] P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3), 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. [32] Q. Hu and Z. Yin, Blowup phenomena for a new periodic nonlinearly dispersive wave equation, Math. Nachr., 283 (2010), 1613-1628. doi: 10.1002/mana.200810075. [33] R. Beals, D. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), L1-L4. doi: 10.1088/0266-5611/15/1/001. [34] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [35] 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. [36] R. M. Chen and Y. Liu, Wave-breaking and global existence for a generalized two-component Camassa-Holm system,, Int. Math. Res. Not., 2011 (): 1381. [37] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Spectral Theory and Differential Equations" (Proc. Sympos., Dundee, 1974, dedicated to Konrad Jörgens), Lecture Notes in Mathematics, 448, Springer, Berlin, (1975), 25-70. [38] T. Tao, Low-regularity global solutions to nonlinear dispersive equations, in "Survey in Analysis and Operator Theory" (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, (2002), 19-48. [39] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. [40] Y. Liu and P. Zhang, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, International Mathematics Research Notices, 2010 (): 1981. [41] Z. Popowicz, A 2-component or N=2 supersymmetric Camassa-Holm equation, Physics Letters A, 354 (2006), 110-114. doi: 10.1016/j.physleta.2006.01.027. [42] 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.

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References:
 [1] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 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, Analysis and its Applications (Singap.), 5 (2007), 1-27. [3] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, Journal of Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [4] A. Constantin, On the blow-up of solutions of a periodic shallow water equation, Journal of Nonlinear Science, 10 (2000), 391-399. doi: 10.1007/s003329910017. [5] A. Constantin, The trajectories of particles in Stokes waves, Inventiones Mathematicae, 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [6] A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynamics Research, 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004. [7] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [8] A. Constantin and J. Escher, Well-posedness, global existence, and blowup 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. [9] 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. [10] A. Constantin and H. P. Mckean, A shallow water equation on the circle, Communications on Pure and Applied Mathematics, 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [11] A. Constantin and W. A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics, 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. [12] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, Journal of Nonlinear Science, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. [13] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D., 4 (): 47. [14] A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation, Differential Integral Equations, 14 (2001), 821-831. [15] A. Shabat and L. Martínez Alonso, On the prolongation of a hierarchy of hydrodynamic chains, in "New Trends in Integrability and Partial Solvability," NATO Sci. Ser. II Math. Phys. Chem., 132, Kluwer Acad. Publ., Dordrecht, (2004), 263-280. [16] C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, Journal of Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002. [17] C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [18] 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. [19] G. B. Whitham, "Linear and Nonlinear Waves," Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. [20] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, Journal of Functional Analysis, 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. [21] G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Mathematische Zeitschrift, 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2. [22] G. Misiołek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104. doi: 10.1007/PL00012648. [23] G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [24] H. Aratyn, J. F. Gomes and A. H. Zimerman, On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation, SIGMA Symmetry, Integrability and Geometry: Methods and Applications, 2 (2006), Paper 070, 12 pp. [25] H. P. Mckean, Breakdown of a shallow water equation, Asian Journal of Mathematics, 2 (1998), 867-874. [26] 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. [27] J. F. Toland, Stokes waves, Topological Methods in Nonlinear Analysis, 7 (1996), 1-48. [28] J. Lenells, Stability of periodic peakons,, International Mathematics Research Notices, 2004 (): 485. [29] M. Chen, S.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Letters in Mathematical Physics, 75 (2006), 1-15. doi: 10.1007/s11005-005-0041-7. [30] O. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system, Wave Motion, 46 (2009), 397-402. doi: 10.1016/j.wavemoti.2009.06.011. [31] P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3), 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. [32] Q. Hu and Z. Yin, Blowup phenomena for a new periodic nonlinearly dispersive wave equation, Math. Nachr., 283 (2010), 1613-1628. doi: 10.1002/mana.200810075. [33] R. Beals, D. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), L1-L4. doi: 10.1088/0266-5611/15/1/001. [34] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [35] 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. [36] R. M. Chen and Y. Liu, Wave-breaking and global existence for a generalized two-component Camassa-Holm system,, Int. Math. Res. Not., 2011 (): 1381. [37] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Spectral Theory and Differential Equations" (Proc. Sympos., Dundee, 1974, dedicated to Konrad Jörgens), Lecture Notes in Mathematics, 448, Springer, Berlin, (1975), 25-70. [38] T. Tao, Low-regularity global solutions to nonlinear dispersive equations, in "Survey in Analysis and Operator Theory" (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, (2002), 19-48. [39] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. [40] Y. Liu and P. Zhang, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, International Mathematics Research Notices, 2010 (): 1981. [41] Z. Popowicz, A 2-component or N=2 supersymmetric Camassa-Holm equation, Physics Letters A, 354 (2006), 110-114. doi: 10.1016/j.physleta.2006.01.027. [42] 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.
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