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

Shouming Zhou; Chunlai Mu

doi:10.3934/dcds.2013.33.1713

Article Contents

2013, Volume 33, Issue 4: 1713-1739. Doi: 10.3934/dcds.2013.33.1713

This issue Previous Article Well-posedness for a modified two-component Camassa-Holm system in critical spaces Next Article

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

Received: October 31, 2011

Revised: January 31, 2012

Published: March 2013

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35B10, 35Q53.

Citation:

Related Papers

Cited by

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.