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A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity
Blow-up for the two-component Camassa--Holm system
1. | Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim |
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, New York, 2000. |
[2] |
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. |
[3] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
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. |
[5] |
R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Inter. Math Research Notices, (2011), 1381-1416.
doi: 10.1093/imrn/rnq118. |
[6] |
A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Physics Letters A, 372 (2008), 7129-7132.
doi: 10.1016/j.physleta.2008.10.050. |
[7] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[8] |
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.
doi: 10.3934/dcds.2007.19.493. |
[9] |
Y. Fu and C. Qu, Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 012906, 25pp.
doi: 10.1063/1.3064810. |
[10] |
K. Grunert, H. Holden and X. Raynaud, Global solutions for the two-component Camassa-Holm system, Comm. Partial Differential Equations, 37 (2012), 2245-2271.
doi: 10.1080/03605302.2012.683505. |
[11] |
K. Grunert, H. Holden and X. Raynaud, Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics, Nonlinear Anal. Real World Appl., 17 (2014), 203-244.
doi: 10.1016/j.nonrwa.2013.12.001. |
[12] |
K. Grunert, H. Holden and X. Raynaud, A continuous interpolation between conservative and dissipative solutions for the Camassa-Holm system, arXiv:1402.1060. |
[13] |
C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena (eds. H. Holden and K. H. Karlsen), Amer. Math. Soc., Providence, 526 (2010), 199-220.
doi: 10.1090/conm/526/10382. |
[14] |
C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. I. H. Poincaré - AN, 28 (2011), 623-641.
doi: 10.1016/j.anihpc.2011.04.003. |
[15] |
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.
doi: 10.1016/j.jde.2009.08.002. |
[16] |
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. |
[17] |
G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66.
doi: 10.1007/s00209-009-0660-2. |
[18] |
Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. Math., 124 (2010), 307-322.
doi: 10.1111/j.1467-9590.2009.00472.x. |
[19] |
D. Henry, Infnite 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. |
[20] |
H. Holden and X. Raynaud, Global conservative solutions for the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[21] |
H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.
doi: 10.3934/dcds.2009.24.1047. |
[22] |
Q. Hu, Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system, Applicable Analysis, 92 (2013), 398-410.
doi: 10.1080/00036811.2011.621893. |
[23] |
P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation, Math. Notes, 81 (2007), 130-134.
doi: 10.1134/S0001434607010142. |
[24] |
W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system, J. Math. Phys., 52 (2011), 033507, 24pp.
doi: 10.1063/1.3562928. |
[25] |
M. Yuen, Perturbational blowup solutions to the 2-component Camassa-Holm equations, J. Math. Anal. Appl., 390 (2012), 596-602.
doi: 10.1016/j.jmaa.2011.05.016. |
[26] |
P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021.
doi: 10.1093/imrn/rnp211. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, New York, 2000. |
[2] |
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. |
[3] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
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. |
[5] |
R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Inter. Math Research Notices, (2011), 1381-1416.
doi: 10.1093/imrn/rnq118. |
[6] |
A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Physics Letters A, 372 (2008), 7129-7132.
doi: 10.1016/j.physleta.2008.10.050. |
[7] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[8] |
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.
doi: 10.3934/dcds.2007.19.493. |
[9] |
Y. Fu and C. Qu, Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 012906, 25pp.
doi: 10.1063/1.3064810. |
[10] |
K. Grunert, H. Holden and X. Raynaud, Global solutions for the two-component Camassa-Holm system, Comm. Partial Differential Equations, 37 (2012), 2245-2271.
doi: 10.1080/03605302.2012.683505. |
[11] |
K. Grunert, H. Holden and X. Raynaud, Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics, Nonlinear Anal. Real World Appl., 17 (2014), 203-244.
doi: 10.1016/j.nonrwa.2013.12.001. |
[12] |
K. Grunert, H. Holden and X. Raynaud, A continuous interpolation between conservative and dissipative solutions for the Camassa-Holm system, arXiv:1402.1060. |
[13] |
C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena (eds. H. Holden and K. H. Karlsen), Amer. Math. Soc., Providence, 526 (2010), 199-220.
doi: 10.1090/conm/526/10382. |
[14] |
C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. I. H. Poincaré - AN, 28 (2011), 623-641.
doi: 10.1016/j.anihpc.2011.04.003. |
[15] |
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.
doi: 10.1016/j.jde.2009.08.002. |
[16] |
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. |
[17] |
G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66.
doi: 10.1007/s00209-009-0660-2. |
[18] |
Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. Math., 124 (2010), 307-322.
doi: 10.1111/j.1467-9590.2009.00472.x. |
[19] |
D. Henry, Infnite 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. |
[20] |
H. Holden and X. Raynaud, Global conservative solutions for the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[21] |
H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.
doi: 10.3934/dcds.2009.24.1047. |
[22] |
Q. Hu, Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system, Applicable Analysis, 92 (2013), 398-410.
doi: 10.1080/00036811.2011.621893. |
[23] |
P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation, Math. Notes, 81 (2007), 130-134.
doi: 10.1134/S0001434607010142. |
[24] |
W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system, J. Math. Phys., 52 (2011), 033507, 24pp.
doi: 10.1063/1.3562928. |
[25] |
M. Yuen, Perturbational blowup solutions to the 2-component Camassa-Holm equations, J. Math. Anal. Appl., 390 (2012), 596-602.
doi: 10.1016/j.jmaa.2011.05.016. |
[26] |
P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021.
doi: 10.1093/imrn/rnp211. |
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