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July  2013, 33(7): 2809-2827. doi: 10.3934/dcds.2013.33.2809

Lipschitz metric for the Camassa--Holm equation on the line

Katrin Grunert 1, , Helge Holden 1, and  Xavier Raynaud 2,

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim
2. 

Centre of Mathematics for Applications, University of Oslo, NO-0316 Oslo

Received  March 2012 Revised  May 2012 Published  January 2013

We study stability of solutions of the Cauchy problem on the line for the Camassa--Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_D(u(t),v(t))\le e^{Ct} d_D(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f''(u)(u_x)^2)_x=0$ (for $f$ without inflection points).
Keywords: conservative solutions., Camassa--Holm equation, Lipschitz metric.
Mathematics Subject Classification: Primary: 35Q53, 35B35; Secondary: 35Q2.
Citation: Katrin Grunert, Helge Holden, Xavier Raynaud. Lipschitz metric for the Camassa--Holm equation on the line. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2809-2827. doi: 10.3934/dcds.2013.33.2809
References:
[1]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar

[2]

A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl., 94 (2010), 68-92. doi: 10.1016/j.matpur.2010.02.005.  Google Scholar

[3]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[4]

R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[5]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328.  Google Scholar

[6]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[7]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[8]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[9]

H.-H. Dai, Exact traveling-wave solutions of an integrable equation arising in hyperelastic rods, Wave Motion, 28 (1998), 367-381. doi: 10.1016/S0165-2125(98)00014-6.  Google Scholar

[10]

H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373.  Google Scholar

[11]

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

[12]

K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation, J. Differential Equations, 250 (2011), 1460-1492. doi: 10.1016/j.jde.2010.07.006.  Google Scholar

[13]

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. doi: 10.1080/03605300601088674.  Google Scholar

[14]

H. Holden and X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differ. Equ., 4 (2007), 39-64. doi: 10.1142/S0219891607001045.  Google Scholar

[15]

H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. doi: 10.1016/j.jde.2006.09.007.  Google Scholar

show all references

References:
[1]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar

[2]

A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl., 94 (2010), 68-92. doi: 10.1016/j.matpur.2010.02.005.  Google Scholar

[3]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[4]

R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[5]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328.  Google Scholar

[6]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[7]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[8]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[9]

H.-H. Dai, Exact traveling-wave solutions of an integrable equation arising in hyperelastic rods, Wave Motion, 28 (1998), 367-381. doi: 10.1016/S0165-2125(98)00014-6.  Google Scholar

[10]

H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373.  Google Scholar

[11]

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

[12]

K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation, J. Differential Equations, 250 (2011), 1460-1492. doi: 10.1016/j.jde.2010.07.006.  Google Scholar

[13]

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. doi: 10.1080/03605300601088674.  Google Scholar

[14]

H. Holden and X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differ. Equ., 4 (2007), 39-64. doi: 10.1142/S0219891607001045.  Google Scholar

[15]

H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. doi: 10.1016/j.jde.2006.09.007.  Google Scholar

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