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

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

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.
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
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show all references

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

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

[3]

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

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

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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

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

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

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

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

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

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

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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

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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]

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[25]

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[26]

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[28]

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

[29]

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

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