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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 and Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
References:
[1]

R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239.

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[4]

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.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London(A), 457 (2001), 953-970.

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.

[7]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.

[8]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[9]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069.

[10]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.

[11]

A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.

[12]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.

[13]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.

[14]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.

[15]

H. H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363. doi: 10.1098/rspa.2000.0520.

[16]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66.

[17]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007) 1511-1549.

[18]

H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484.

[19]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.

[20]

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

[21]

O. G. Mustafa, On the Cauchy problem for a generalized Camassa-Holm equation, Nonlinear Anal. TMA, 64 (2006), 1382-1399.

[22]

O. G. Mustafa, Solitary waves for a generalized Camassa-Holm equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 205-212.

[23]

O. G. Mustafa, Global conservative solutions of the hyperelastic rod equation, Int. Math. Res. Notices, (2007), Art. ID rnm 040, 26 pp.

[24]

O. G. Mustafa, Global dissipative solution of the generalized Camassa-Holm equation, J. Nonlinear Math. Phys., 15 (2008), 96-115.

[25]

L. Tian and X. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19 (2004), 621-637.

[26]

J. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26 (2005), 1149-1162.

[27]

Z. Yin, On the Cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys., 10 (2003), 10-15. doi: 10.2991/jnmp.2003.10.1.2.

[28]

Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Anal. TMA, 66 (2007), 460-471.

[29]

Z. Yin, On the blow-up scenario for the generalized Camassa-Holm equation, Comm. Partial Differential Equations, 29 (2004), 867-877.

show all references

References:
[1]

R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239.

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[4]

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.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London(A), 457 (2001), 953-970.

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.

[7]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.

[8]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[9]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069.

[10]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.

[11]

A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.

[12]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.

[13]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.

[14]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.

[15]

H. H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363. doi: 10.1098/rspa.2000.0520.

[16]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66.

[17]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007) 1511-1549.

[18]

H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484.

[19]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.

[20]

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

[21]

O. G. Mustafa, On the Cauchy problem for a generalized Camassa-Holm equation, Nonlinear Anal. TMA, 64 (2006), 1382-1399.

[22]

O. G. Mustafa, Solitary waves for a generalized Camassa-Holm equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 205-212.

[23]

O. G. Mustafa, Global conservative solutions of the hyperelastic rod equation, Int. Math. Res. Notices, (2007), Art. ID rnm 040, 26 pp.

[24]

O. G. Mustafa, Global dissipative solution of the generalized Camassa-Holm equation, J. Nonlinear Math. Phys., 15 (2008), 96-115.

[25]

L. Tian and X. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19 (2004), 621-637.

[26]

J. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26 (2005), 1149-1162.

[27]

Z. Yin, On the Cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys., 10 (2003), 10-15. doi: 10.2991/jnmp.2003.10.1.2.

[28]

Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Anal. TMA, 66 (2007), 460-471.

[29]

Z. Yin, On the blow-up scenario for the generalized Camassa-Holm equation, Comm. Partial Differential Equations, 29 (2004), 867-877.

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