American Institute of Mathematical Sciences

Advanced
April  2013, 33(4): 1713-1739. doi: 10.3934/dcds.2013.33.1713

Global conservative and dissipative solutions of the generalized Camassa-Holm equation

Shouming Zhou 1, and  Chunlai Mu 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China, China

Received  November 2011 Revised  February 2012 Published  October 2012

This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equation beyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.
Keywords: dissipative solutions., conservative solutions, Generalized Camassa-Holm equation.
Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35B10, 35Q5.
Citation: 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
References:
[1]

Adv. Math., 140 (1998), 190-206.  Google Scholar

[2]

Arch. Rational Mech. Anal., 183 (2007), 215-239.  Google Scholar

[3]

Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar

[4]

C. R. Math. Acad. Sci. Paris, 343 (2006), 627-632.  Google Scholar

[5]

Proc. Roy. Soc. London(A), 457 (2001), 953-970.  Google Scholar

[6]

Invent. Math., 166 (2006), 523-535.  Google Scholar

[7]

Phys. Rev. Lett., 71 (1993), 1661-1664.  Google Scholar

[8]

Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[9]

SIAM J. Math. Anal., 37 (2005), 1044-1069.  Google Scholar

[10]

Ann. of Math., 173 (2011), 559-568.  Google Scholar

[11]

Inverse Problems, 22 (2006), 2197-2207.  Google Scholar

[12]

Comm. Pure Appl. Math., 53 (2000), 603-610.  Google Scholar

[13]

Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.  Google Scholar

[14]

J. Nonlinear. Sci., 12 (2002), 415-422.  Google Scholar

[15]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363. doi: 10.1098/rspa.2000.0520.  Google Scholar

[16]

Phys. D, 4 (1981), 47-66.  Google Scholar

[17]

Comm. Partial Differential Equations, 32 (2007) 1511-1549.  Google Scholar

[18]

J. Differential Equations, 233 (2007), 448-484.  Google Scholar

[19]

Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.  Google Scholar

[20]

J. Phys. A, 38 (2005), 869-880. doi: 10.1088/0305-4470/38/4/007.  Google Scholar

[21]

Nonlinear Anal. TMA, 64 (2006), 1382-1399.  Google Scholar

[22]

Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 205-212.  Google Scholar

[23]

Int. Math. Res. Notices, (2007), Art. ID rnm 040, 26 pp.  Google Scholar

[24]

J. Nonlinear Math. Phys., 15 (2008), 96-115.  Google Scholar

[25]

Chaos Solitons Fractals, 19 (2004), 621-637.  Google Scholar

[26]

Chaos Solitons Fractals, 26 (2005), 1149-1162.  Google Scholar

[27]

J. Nonlinear Math. Phys., 10 (2003), 10-15. doi: 10.2991/jnmp.2003.10.1.2.  Google Scholar

[28]

Nonlinear Anal. TMA, 66 (2007), 460-471.  Google Scholar

[29]

Comm. Partial Differential Equations, 29 (2004), 867-877.  Google Scholar

show all references

References:
[1]

Adv. Math., 140 (1998), 190-206.  Google Scholar

[2]

Arch. Rational Mech. Anal., 183 (2007), 215-239.  Google Scholar

[3]

Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar

[4]

C. R. Math. Acad. Sci. Paris, 343 (2006), 627-632.  Google Scholar

[5]

Proc. Roy. Soc. London(A), 457 (2001), 953-970.  Google Scholar

[6]

Invent. Math., 166 (2006), 523-535.  Google Scholar

[7]

Phys. Rev. Lett., 71 (1993), 1661-1664.  Google Scholar

[8]

Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[9]

SIAM J. Math. Anal., 37 (2005), 1044-1069.  Google Scholar

[10]

Ann. of Math., 173 (2011), 559-568.  Google Scholar

[11]

Inverse Problems, 22 (2006), 2197-2207.  Google Scholar

[12]

Comm. Pure Appl. Math., 53 (2000), 603-610.  Google Scholar

[13]

Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.  Google Scholar

[14]

J. Nonlinear. Sci., 12 (2002), 415-422.  Google Scholar

[15]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363. doi: 10.1098/rspa.2000.0520.  Google Scholar

[16]

Phys. D, 4 (1981), 47-66.  Google Scholar

[17]

Comm. Partial Differential Equations, 32 (2007) 1511-1549.  Google Scholar

[18]

J. Differential Equations, 233 (2007), 448-484.  Google Scholar

[19]

Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.  Google Scholar

[20]

J. Phys. A, 38 (2005), 869-880. doi: 10.1088/0305-4470/38/4/007.  Google Scholar

[21]

Nonlinear Anal. TMA, 64 (2006), 1382-1399.  Google Scholar

[22]

Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 205-212.  Google Scholar

[23]

Int. Math. Res. Notices, (2007), Art. ID rnm 040, 26 pp.  Google Scholar

[24]

J. Nonlinear Math. Phys., 15 (2008), 96-115.  Google Scholar

[25]

Chaos Solitons Fractals, 19 (2004), 621-637.  Google Scholar

[26]

Chaos Solitons Fractals, 26 (2005), 1149-1162.  Google Scholar

[27]

J. Nonlinear Math. Phys., 10 (2003), 10-15. doi: 10.2991/jnmp.2003.10.1.2.  Google Scholar

[28]

Nonlinear Anal. TMA, 66 (2007), 460-471.  Google Scholar

[29]

Comm. Partial Differential Equations, 29 (2004), 867-877.  Google Scholar

[1]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029
[2]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065
[3]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[4]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[5]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[6]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[7]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005
[8]

Huan Zhang, Jun Zhou. Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021034
[9]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016
[10]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033
[11]

Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061
[12]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
[13]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[14]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
[15]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047
[16]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, 2021, 14 (2) : 389-406. doi: 10.3934/krm.2021009
[17]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017
[18]

Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021098
[19]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056
[20]

Shixiong Wang, Longjiang Qu, Chao Li, Shaojing Fu, Hao Chen. Finding small solutions of the equation $ \mathit{{Bx-Ay = z}} $ and its applications to cryptanalysis of the RSA cryptosystem. Advances in Mathematics of Communications, 2021, 15 (3) : 441-469. doi: 10.3934/amc.2020076

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences