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Blow-up criteria of solutions to a modified two-component Camassa-Holm system. (English) Zbl 1231.35200
Summary: We consider a modified two-component Camassa-Holm (MCH2) system which arises in shallow water theory. We analyze the wave breaking mechanism by establishing some new blow-up criteria for this system formulated either on the line or with space-periodic initial condition.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35B44Blow-up (PDE)
35Q35PDEs in connection with fluid mechanics
References:
[1]Olver, P.; Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. rev. E 53, 1900-1906 (1996)
[2]Constantin, A.; Ivanov, R.: On an integrable two-component Camassa–Holm shallow water system, Phys. lett. A 372, 7129-7132 (2008) · Zbl 1227.76016 · doi:10.1016/j.physleta.2008.10.050
[3]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 · doi:10.1007/s11005-005-0041-7
[4]Falqui, G.: On a Camassa–Holm type equation with two dependent variables, J. phys. A 39, 327-342 (2006) · Zbl 1084.37053 · doi:10.1088/0305-4470/39/2/004
[5]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
[6]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 · doi:10.1016/j.jfa.2010.02.008
[7]Guo, Z.; Zhou, Y.: On solutions to a two-component generalized Camassa–Holm equation, Stud. appl. Math. 124, 307-322 (2010) · Zbl 1189.35255 · doi:10.1111/j.1467-9590.2009.00472.x
[8]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 · doi:10.1016/j.jmaa.2010.06.046
[9]Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solitons, Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[10]Bressan, A.; Constantin, A.: Global conservative solutions of the Camassa–Holm equation, Arch. ration. Mech. anal. 183, 215-239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[11]Constantin, A.; Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Commun. pure appl. Math. 51, 475-504 (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[12]Guo, Z.: Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa–Holm equation, J. math. Phys. 49, 033516 (2008) · Zbl 1153.81368 · doi:10.1063/1.2885075
[13]Himonas, A.; Misiołek, G.; Ponce, G.; Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation, Commun. math. Phys. 271, 511-522 (2007) · Zbl 1142.35078 · doi:10.1007/s00220-006-0172-4
[14]Mckean, H. P.: Breakdown of a shallow water equation, Asian J. Math. 2, 767-774 (1998) · Zbl 0959.35140 · doi:http://www.intlpress.com/AJM/p/1998/2_4/AJM-2-4-867-874.pdf
[15]Zhou, Y.: Wave breaking for a shallow water equation, Nonlinear anal. 57, 137-152 (2004) · Zbl 1106.35070 · doi:10.1016/j.na.2004.02.004
[16]Zhou, Y.: Wave breaking for a periodic shallow water equation, J. math. Anal. appl. 290, 591-604 (2004) · Zbl 1042.35060 · doi:10.1016/j.jmaa.2003.10.017
[17]Xin, Z.; Zhang, P.: On the weak solutions to a shallow water equation, Commun. pure appl. Math. 53, 1411-1433 (2000) · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[18]Constantin, A.; Strauss, W.: Stability of peakons, Commun. pure appl. Math. 53, 603-610 (2000) · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[19]Zhou, Y.: Stability of solitary waves for a rod equation, Chaos solitons fractals 21, 977-981 (2004) · Zbl 1046.35094 · doi:10.1016/j.chaos.2003.12.030
[20]Henry, D.: Infinite propagation speed for a two component Camassa–Holm equation, Discrete contin. Dyn. syst. Ser. B 12, 597-606 (2009) · Zbl 1180.35458 · doi:10.3934/dcdsb.2009.12.597
[21]Mustafa, O. G.: On smooth traveling waves of an integrable two-component Camassa–Holm shallow water system, Wave motion 46, 397-402 (2009) · Zbl 1231.76063 · doi:10.1016/j.wavemoti.2009.06.011
[22]Zhang, P.; Liu, Y.: Stability of solitary waves and wave-breaking phenomena for the two-component Camassa–Holm system, Int. math. Res. not., 1981-2021 (2010)
[23]Jin, L.; Guo, Z.: On a two-component Degasperis–Procesi shallow water system, Nonlinear analysis RWA 11, 4164-4173 (2010)
[24]Ni, L.: The Cauchy problem for a two-component generalized θ-equations, Nonlinear anal. 73, 1338-1349 (2010)
[25]Wunsch, M.: The generalized Hunter–Saxton system, SIAM J. Math. anal. 42, 1286-1304 (2010) · Zbl 1223.35092 · doi:10.1137/090768576
[26]Holm, D.; Náraigh, L. Ó.; Tronci, C.: Singular solutions of a modified two-component Camassa–Holm equation, Phys. rev. E 3, No. 79, 016601 (2009)
[27]Fu, Y.; Liu, Y.; Qu, C.: Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons, Math. ann. 348, 415-448 (2010) · Zbl 1207.35074 · doi:10.1007/s00208-010-0483-9
[28]Guan, C.; Karlsen, K. H.; Yin, Z.: Well-posedness and blow-up phenomena for a modified two-component Camassa–Holm equation, nonlinear partial differential equations and hyperbolic wave phenomena, Contemp. math. 526, 199-220 (2010) · Zbl 1213.35133
[29]Marsden, J.; Ratiu, T.: Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in applied mathematics 17 (1999)
[30]Holm, D.; Marsden, J.; Ratiu, T.: The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv. math. 137, 1-81 (1998) · Zbl 0951.37020 · doi:10.1006/aima.1998.1721
[31]Kato, T.: Quasi-linear equations of evolution with application to partial differential equations, Lecture notes in math. 448, 25-70 (1975) · Zbl 0315.35077
[32]Zhou, Y.: Blow-up of solutions to a nonlinear dispersive rod equation, Calc. var. Partial differential equations 25, 63-77 (2006) · Zbl 1172.35504 · doi:10.1007/s00526-005-0358-1