American Institute of Mathematical Sciences

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March  2021, 29(1): 1691-1708. doi: 10.3934/era.2020087

Global conservative solutions for a modified periodic coupled Camassa-Holm system

Rong Chen 1, , Shihang Pan 2,, and  Baoshuai Zhang 3,,

1. 

Personnel Department, Chongqing Normal University, Chongqing 401331, China
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3. 

School of Economic Management, Chongqing Normal University, Chongqing 401331, China

* Corresponding author: The first corresponding author Baoshuai Zhang and the second corresponding author Shihang Pan

Received  April 2020 Revised  July 2020 Published  March 2021 Early access  August 2020

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.

Keywords: Periodic coupled Camassa-Holm system, global conservative solution.
Mathematics Subject Classification: Primary: 35A01, 35A02; Secondary: 35D30, 35G25, 35G25.
Citation: Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087
References:
[1]

R. Beals, D. H. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206.  doi: 10.1006/aima.1998.1768.  Google Scholar

[2]

A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris, 343 (2006), 627-632.  doi: 10.1016/j.crma.2006.10.014.  Google Scholar

[3]

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[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.  Google Scholar

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A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328.   Google Scholar

[8]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1545.  doi: 10.1512/iumj.1998.47.1466.  Google Scholar

[9]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[10]

A. Constantin and R. I. 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.  Google Scholar

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A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.  Google Scholar

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Y. Fu and C. Qu, Well-possdness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 677-702.  doi: 10.1063/1.3064810.  Google Scholar

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Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modifed two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448.  doi: 10.1007/s00208-010-0483-9.  Google Scholar

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C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 28 (2011), 623-641.  doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

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C. Guan, K. Yan and X. Wei, Lipschitz metric for the modified two-component Camassa-Holm system, Anal. Appl. (Singap.), 16 (2018), 159-182.  doi: 10.1142/S0219530516500226.  Google Scholar

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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.  Google Scholar

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H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier (Grenoble), 58 (2008), 945-988.  doi: 10.5802/aif.2375.  Google Scholar

[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.  Google Scholar

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J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.  doi: 10.1088/0305-4470/38/4/007.  Google Scholar

[25]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[26]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J. Funct. Anal., 261 (2011), 1204-1226.  doi: 10.1016/j.jfa.2011.04.015.  Google Scholar

[27]

W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582.  doi: 10.1016/j.jde.2011.08.010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar

[33]

S. Zhou, Non-uniform dependence and persistence properties for coupled Camassa-Holm equations, Math. Methods Appl. Sci., 40 (2017), 3718-3732.  doi: 10.1002/mma.4258.  Google Scholar

[34]

S. Zhou, Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747.  doi: 10.1007/s00028-014-0236-4.  Google Scholar

[35]

S. Zhou, Z. Qiao, C. Mu and L. Wei, Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude, J. Differential Equations, 263 (2017), 910-933.  doi: 10.1016/j.jde.2017.03.002.  Google Scholar

show all references

References:
[1]

R. Beals, D. H. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206.  doi: 10.1006/aima.1998.1768.  Google Scholar

[2]

A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris, 343 (2006), 627-632.  doi: 10.1016/j.crma.2006.10.014.  Google Scholar

[3]

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.  Google Scholar

[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.  Google Scholar

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328.   Google Scholar

[8]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1545.  doi: 10.1512/iumj.1998.47.1466.  Google Scholar

[9]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[10]

A. Constantin and R. I. 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.  Google Scholar

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[12]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.  Google Scholar

[13]

Y. Fu and C. Qu, Well-possdness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 677-702.  doi: 10.1063/1.3064810.  Google Scholar

[14]

Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modifed two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448.  doi: 10.1007/s00208-010-0483-9.  Google Scholar

[15]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[16]

C. Guan, Uniqueness of global conservative weak solutions for the modified two-component Camassa-Holm system, J. Evol. Equ., 18 (2018), 1003-1024.  doi: 10.1007/s00028-018-0430-x.  Google Scholar

[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.  Google Scholar

[18]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 28 (2011), 623-641.  doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

[19]

C. Guan and Z. Yin, On the global weak solutions for a modified two-component Camassa-Holm equation, Math. Nachr., 286 (2013), 1287-1304.  doi: 10.1002/mana.201200193.  Google Scholar

[20]

C. Guan, K. Yan and X. Wei, Lipschitz metric for the modified two-component Camassa-Holm system, Anal. Appl. (Singap.), 16 (2018), 159-182.  doi: 10.1142/S0219530516500226.  Google Scholar

[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.  Google Scholar

[22]

H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier (Grenoble), 58 (2008), 945-988.  doi: 10.5802/aif.2375.  Google Scholar

[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.  Google Scholar

[24]

J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.  doi: 10.1088/0305-4470/38/4/007.  Google Scholar

[25]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[26]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J. Funct. Anal., 261 (2011), 1204-1226.  doi: 10.1016/j.jfa.2011.04.015.  Google Scholar

[27]

W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582.  doi: 10.1016/j.jde.2011.08.010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar

[33]

S. Zhou, Non-uniform dependence and persistence properties for coupled Camassa-Holm equations, Math. Methods Appl. Sci., 40 (2017), 3718-3732.  doi: 10.1002/mma.4258.  Google Scholar

[34]

S. Zhou, Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747.  doi: 10.1007/s00028-014-0236-4.  Google Scholar

[35]

S. Zhou, Z. Qiao, C. Mu and L. Wei, Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude, J. Differential Equations, 263 (2017), 910-933.  doi: 10.1016/j.jde.2017.03.002.  Google Scholar

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