American Institute of Mathematical Sciences

Advanced
January  2011, 10(1): 45-57. doi: 10.3934/cpaa.2011.10.45

The existence of weak solutions for a generalized Camassa-Holm equation

Shaoyong Lai 1, , Qichang Xie 2, , Yunxi Guo 2, and  YongHong Wu 2,

1. 

Department of Mathematics,Sichuan Normal University, Chengdu, Department of Mathematics and Statistics,Curtin University of Technology, Perth, China
2. 

Department of Applied Mathematics, Southwestern University of Finance and Economics, 610074, Chengdu, China, China, China

Received  October 2009 Revised  August 2010 Published  November 2010

A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.
Keywords: high order nonlinear terms, pseudoparabolic regularization technique., weak solution, Generalized Camassa-Holm equation.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35L0.
Citation: Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45
References:
[1]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: doi:10.1103/PhysRevLett.71.1661.  Google Scholar

[2]

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

[3]

Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation, Discrete and Continuous Dynamical Systems, 21 (2008), 883-906. doi: doi:10.3934/dcds.2008.21.883.  Google Scholar

[4]

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

[5]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: doi:10.1016/S0022-0396(03)00096-2.  Google Scholar

[6]

A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions, Appl. Math. Comput., 165 (2005), 485-501. doi: doi:10.1016/j.amc.2004.04.029.  Google Scholar

[7]

A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195 (2008), 24-33. doi: doi:10.1016/j.amc.2007.04.066.  Google Scholar

[8]

L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos, Solitons and Fractals, 19 (2004), 621-637. doi: doi:10.1016/S0960-0779(03)00192-9.  Google Scholar

[9]

Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations, Phys. Lett. A, 372 (2008), 4141-4143. doi: doi:10.1016/j.physleta.2007.03.096.  Google Scholar

[10]

S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation, Communications in Partial Differential Equations, 30 (2005), 761-781. doi: doi:10.1081/PDE-200059284.  Google Scholar

[11]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: doi:10.1002/cpa.3160410704.  Google Scholar

[12]

J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: doi:10.1098/rsta.1975.0035.  Google Scholar

[13]

A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$, Communications on Pure and Applied Analysis, 7 (2008), 89-105. doi: doi:10.3934/cpaa.2008.7.89.  Google Scholar

[14]

R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Communications on Pure and Applied Analysis, 7 (2008), 1077-1090. doi: doi:10.3934/cpaa.2008.7.1077.  Google Scholar

[15]

D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Communications on Pure and Applied Analysis, 7 (2008), 867-881. doi: doi:10.3934/cpaa.2008.7.867.  Google Scholar

[16]

K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Communications on Pure and Applied Analysis, 7 (2008), 1203-1210. doi: doi:10.3934/cpaa.2008.7.1203.  Google Scholar

[17]

C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow, J. Differential Equations, 246 (2009), 373-390. doi: doi:10.1016/j.jde.2008.06.026.  Google Scholar

show all references

References:
[1]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: doi:10.1103/PhysRevLett.71.1661.  Google Scholar

[2]

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

[3]

Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation, Discrete and Continuous Dynamical Systems, 21 (2008), 883-906. doi: doi:10.3934/dcds.2008.21.883.  Google Scholar

[4]

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

[5]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: doi:10.1016/S0022-0396(03)00096-2.  Google Scholar

[6]

A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions, Appl. Math. Comput., 165 (2005), 485-501. doi: doi:10.1016/j.amc.2004.04.029.  Google Scholar

[7]

A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195 (2008), 24-33. doi: doi:10.1016/j.amc.2007.04.066.  Google Scholar

[8]

L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos, Solitons and Fractals, 19 (2004), 621-637. doi: doi:10.1016/S0960-0779(03)00192-9.  Google Scholar

[9]

Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations, Phys. Lett. A, 372 (2008), 4141-4143. doi: doi:10.1016/j.physleta.2007.03.096.  Google Scholar

[10]

S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation, Communications in Partial Differential Equations, 30 (2005), 761-781. doi: doi:10.1081/PDE-200059284.  Google Scholar

[11]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: doi:10.1002/cpa.3160410704.  Google Scholar

[12]

J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: doi:10.1098/rsta.1975.0035.  Google Scholar

[13]

A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$, Communications on Pure and Applied Analysis, 7 (2008), 89-105. doi: doi:10.3934/cpaa.2008.7.89.  Google Scholar

[14]

R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Communications on Pure and Applied Analysis, 7 (2008), 1077-1090. doi: doi:10.3934/cpaa.2008.7.1077.  Google Scholar

[15]

D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Communications on Pure and Applied Analysis, 7 (2008), 867-881. doi: doi:10.3934/cpaa.2008.7.867.  Google Scholar

[16]

K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Communications on Pure and Applied Analysis, 7 (2008), 1203-1210. doi: doi:10.3934/cpaa.2008.7.1203.  Google Scholar

[17]

C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow, J. Differential Equations, 246 (2009), 373-390. doi: doi:10.1016/j.jde.2008.06.026.  Google Scholar

[1]

Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[2]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
[3]

Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243
[4]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
[5]

Wenxia Chen, Jingyi Liu, Danping Ding, Lixin Tian. Blow-up for two-component Camassa-Holm equation with generalized weak dissipation. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3769-3784. doi: 10.3934/cpaa.2020166
[6]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[7]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483
[8]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[9]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
[10]

Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[11]

Li Yang, Chunlai Mu, Shouming Zhou, Xinyu Tu. The global conservative solutions for the generalized camassa-holm equation. Electronic Research Archive, 2019, 27: 37-67. doi: 10.3934/era.2019009
[12]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[13]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[14]

Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029
[15]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[16]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065
[17]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
[18]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159
[19]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501
[20]

Xingxing Liu, Zhijun Qiao, Zhaoyang Yin. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1283-1304. doi: 10.3934/cpaa.2014.13.1283

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences