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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 2809 - 2827, Volume 33, Issue 7, July 2013      doi:10.3934/dcds.2013.33.2809

 
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Katrin Grunert - Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (email)
Helge Holden - Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (email)
Xavier Raynaud - Centre of Mathematics for Applications, University of Oslo, NO-0316 Oslo, Norway (email)

Abstract: 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:  Camassa--Holm equation, Lipschitz metric, conservative solutions.
Mathematics Subject Classification:  Primary: 35Q53, 35B35; Secondary: 35Q20.

Received: March 2012;      Revised: May 2012;      Available Online: January 2013.

 References