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Some properties for almost cellular algebras
Global conservative solutions for a modified periodic coupled Camassa-Holm system
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 |
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.
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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[11] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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] |
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. |
[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] |
J. Lenells,
Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.
doi: 10.1088/0305-4470/38/4/007. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[11] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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] |
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. |
[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] |
J. Lenells,
Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.
doi: 10.1088/0305-4470/38/4/007. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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. |
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