In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic Coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.
Citation: Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system[J]. Electronic Research Archive, 2021, 29(1): 1691-1708. doi: 10.3934/era.2020087
In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic Coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.
[1] |
Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. (1998) 140: 190-206. ![]() |
[2] |
Riemann-Hilbert approach for the Camassa-Holm equation on the line. C. R. Math. Acad. Sci. Paris (2006) 343: 627-632. ![]() |
[3] |
Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. (2007) 183: 215-239. ![]() |
[4] |
An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. (1993) 71: 1661-1664. ![]() |
[5] |
On the scattering problem for the Camassa-Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. (2001) 457: 953-970. ![]() |
[6] |
Existence of permanent and breaking waves for a shallow water equation: A geometric approach. Ann. Inst. Fourier (Grenoble) (2000) 50: 321-362. ![]() |
[7] | Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1998) 26: 303-328. |
[8] |
Global weak solutions for a shallow water equation. Indiana Univ. Math. J. (1998) 47: 1527-1545. ![]() |
[9] |
Inverse scattering transform for the Camassa-Holm equation. Inverse Problems (2006) 22: 2197-2207. ![]() |
[10] |
On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A (2008) 372: 7129-7132. ![]() |
[11] |
Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. (1998) 181: 229-243. ![]() |
[12] |
Global weak solutions for a shallow water equation. Comm. Math. Phys. (2000) 211: 45-61. ![]() |
[13] |
Well-possdness and blow-up solution for a new coupled Camassa-Holm equations with peakons. J. Math. Phys. (2009) 50: 677-702. ![]() |
[14] |
Well-posedness and blow-up solution for a modifed two-component periodic Camassa-Holm system with peakons. Math. Ann. (2010) 348: 415-448. ![]() |
[15] |
Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D (1981/82) 4: 47-66. ![]() |
[16] |
Uniqueness of global conservative weak solutions for the modified two-component Camassa-Holm system. J. Evol. Equ. (2018) 18: 1003-1024. ![]() |
[17] | 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, Contemp. Math., 526, Amer. Math. Soc., Providence, RI, 2010,199–220. |
[18] |
Global weak solutions for a modified two-component Camassa-Holm equation. Ann. Inst. H. Poincaré. Anal. Non Linéaire (2011) 28: 623-641. ![]() |
[19] |
On the global weak solutions for a modified two-component Camassa-Holm equation. Math. Nachr. (2013) 286: 1287-1304. ![]() |
[20] |
Lipschitz metric for the modified two-component Camassa-Holm system. Anal. Appl. (Singap.) (2018) 16: 159-182. ![]() |
[21] |
D. D. Holm, L. Ó Naraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E (3), 79 (2009), 13pp. doi: 10.1103/PhysRevE.79.016601
![]() |
[22] |
Periodic conservative solutions of the Camassa-Holm equation. Ann. Inst. Fourier (Grenoble) (2008) 58: 945-988. ![]() |
[23] |
R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities, Zeitschrift F$\ddot{u}$r Naturforschung A, 61 (2006), 133–138. doi: 10.1515/zna-2006-3-404
![]() |
[24] |
Conservation laws of the Camassa-Holm equation. J. Phys. A (2005) 38: 869-880. ![]() |
[25] |
Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations (2000) 162: 27-63. ![]() |
[26] |
Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation. J. Funct. Anal. (2011) 261: 1204-1226. ![]() |
[27] |
Global conservative solutions of a modified two-component Camassa-Holm shallow water system. J. Differential Equations (2011) 251: 3558-3582. ![]() |
[28] |
L. Tian, Y. Wang and J. Zhou, Global conservative and dissipative solutions of a coupled Camassa-Holm equations, J. Math. Phys., 52 (2011), 29pp. doi: 10.1063/1.3600216
![]() |
[29] |
L. Tian and Y. Xu, Attractor for a viscous coupled Camassa-Holm equation, Adv. Difference Equ., 2010 (2010), 30pp. doi: 10.1155/2010/512812
![]() |
[30] |
L. Tian, W. Yan and G. Gui, On the local well posedness and blow-up solution of a coupled Camassa-Holm equations in Besov spaces, J. Math. Phys., 53 (2012), 10pp. doi: 10.1063/1.3671962
![]() |
[31] |
Y. Wang and Y. Song, Periodic conservative solutions for a modified two-component Camassa-Holm system with Peakons, Abstr. Appl. Anal., 2013 (2013), 12pp. doi: 10.1155/2013/437473
![]() |
[32] |
On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. (2000) 53: 1411-1433. ![]() |
[33] |
Non-uniform dependence and persistence properties for coupled Camassa-Holm equations. Math. Methods Appl. Sci. (2017) 40: 3718-3732. ![]() |
[34] |
Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs. J. Evol. Equ. (2014) 14: 727-747. ![]() |
[35] |
Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude. J. Differential Equations (2017) 263: 910-933. ![]() |