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

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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

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

Received November 2011 Revised February 2012 Published October 2012

Abstract
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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:

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[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
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[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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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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