×

Wave breaking for a modified two-component Camassa-Holm system. (English) Zbl 1230.37093

Summary: We establish sufficient conditions on the initial data to guarantee a blow-up phenomenon for the modified two-component Camassa-Holm (MCH2) system.

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35Q35 PDEs in connection with fluid mechanics
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 215-239 (2007) · Zbl 1105.76013
[2] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5, 1-27 (2007) · Zbl 1139.35378
[3] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[4] Chen, M.; Liu, S.; Zhang, Y., A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75, 1-15 (2006) · Zbl 1105.35102
[5] Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535 (2006) · Zbl 1108.76013
[6] Constantin, A., Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46, 023506 (2005) · Zbl 1076.35109
[7] Constantin, A.; Escher, J., Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173, 559-568 (2011) · Zbl 1228.35076
[8] Constantin, A.; Escher, J., Well-posedness, global existence and blow-up phenomenon for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153
[9] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025
[10] Constantin, A.; Ivanov, R., On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372, 7129-7132 (2008) · Zbl 1227.76016
[11] Constantin, A.; McKean, H. P., A shallow water equation on the circle, Comm. Pure Appl. Math., 52, 949-982 (1999) · Zbl 0940.35177
[12] Constantin, A.; Gerdjikov, V.; Ivanov, R., Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22, 2197-2207 (2006) · Zbl 1105.37044
[13] Constantin, A.; Strauss, W., Stability of peakons, Comm. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149
[14] Escher, J.; Lechtenfeld, O.; Yin, Z., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19, 493-513 (2007) · Zbl 1149.35307
[15] Escher, J.; Kohlmann, M.; Lenells, J., The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, J. Geom. Phys., 61, 436-452 (2011) · Zbl 1210.58007
[16] Falqui, G., On a Camassa-Holm type equation with two dependent variables, J. Phys. A, 39, 327-342 (2006) · Zbl 1084.37053
[17] Fuchssteiner, B.; Fokas, A. S., Symplectic structures, their Bcklund transformations and hereditary symmetries, Phys. D, 4, 1, 4766 (1981/1982) · Zbl 1194.37114
[18] Guan, C.; Karlsen, K. H.; Yin, Z., Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, (Proceedings of the 2008-2009 Special Year in Nonlinear Partial Differential Equations. Proceedings of the 2008-2009 Special Year in Nonlinear Partial Differential Equations, Contemp. Math. (2010), Amer. Math. Soc.), 199-220 · Zbl 1213.35133
[19] Gui, G.; Liu, Y., On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258, 4251-4278 (2010) · Zbl 1189.35254
[20] Guo, Z.; Zhou, Y., On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. Math., 124, 307-322 (2010) · Zbl 1189.35255
[21] Guo, Z., Blow-up and global solutions to a new integrable model with two components, J. Math. Anal. Appl., 372, 316-327 (2010) · Zbl 1205.35045
[22] Guo, Z.; Zhu, M.; Ni, L., Blow-up criteria of solutions to a modified two-component Camassa-Holm system, Nonlinear Anal. Real World Appl., 12, 3531-3540 (2011) · Zbl 1231.35200
[23] Henry, D., Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12, 342-347 (2005) · Zbl 1086.35079
[24] Himonas, A.; Misiolek, G.; Ponce, G.; Zhou, Y., Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271, 511-512 (2007) · Zbl 1142.35078
[25] Holm, D.; Ó Náraigh, L.; Tronci, C., Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E (3), 79, 016601 (2009), 13 pp
[26] Holm, D.; Ivanov, R., Two component CH system: inverse scattering, peakons and geometry, Inverse Problems, 27, 045013 (2011) · Zbl 1216.35175
[27] Ivanov, R., Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A, 365, 2267-2280 (2007) · Zbl 1152.76322
[28] Ionescu-Kruse, D., Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14, 303-312 (2007) · Zbl 1157.76005
[29] Kato, T., (Spectral Theory and Differential Equations, Proc. Sympos.. Spectral Theory and Differential Equations, Proc. Sympos., Dundee, 1974. Spectral Theory and Differential Equations, Proc. Sympos.. Spectral Theory and Differential Equations, Proc. Sympos., Dundee, 1974, Lecture Notes in Math., vol. 48 (1975), Springer-Verlag: Springer-Verlag Berlin), 25, dedicated to Konrad Jorgens
[30] Lenells, J., Conservation laws of the Camassa-Holm equation, J. Phys. A, 38, 869-880 (2005) · Zbl 1076.35100
[31] Li, Y.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119
[32] McKean, H. P., Breakdown of a shallow water equation, Asian J. Math., 2, 767-774 (1998) · Zbl 0959.35140
[33] Olver, P.; Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53, 1900-1906 (1996)
[34] Zhou, Y., Wave breaking for a shallow water equation, Nonlinear Anal., 57, 137-152 (2004) · Zbl 1106.35070
[35] Zhou, Y., Wave breaking for a periodic shallow water equation, J. Math. Anal. Appl., 290, 591-604 (2004) · Zbl 1042.35060
[36] Zhou, Y., Blow-up of solutions to a nonlinear dispersive rod equation, Calc. Var. Partial Differential Equations, 25, 63-77 (2005) · Zbl 1172.35504
[37] Zhou, Y., On solutions to the Holm-Staley b-family of equations, Nonlinearity, 23, 369-381 (2010) · Zbl 1189.37083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.