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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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

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

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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.   Google Scholar

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

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

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

[8]

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

[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.   Google Scholar

[10]

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

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.1098/rspa.2000.0520.  Google Scholar

[16]

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

[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.   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

O. G. Mustafa, Global conservative solutions of the hyperelastic rod equation,, Int. Math. Res. Notices, (2007).   Google Scholar

[24]

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

[25]

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

[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.   Google Scholar

[27]

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

[28]

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

[29]

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

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.   Google Scholar

[2]

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

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.1098/rspa.2000.0520.  Google Scholar

[16]

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

[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.   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

O. G. Mustafa, Global conservative solutions of the hyperelastic rod equation,, Int. Math. Res. Notices, (2007).   Google Scholar

[24]

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

[25]

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

[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.   Google Scholar

[27]

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

[28]

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

[29]

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

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