Lipschitz metric for the CamassaHolm
equation on the line
Katrin Grunert  Department of Mathematical Sciences, Norwegian University of Science and Technology, NO7491 Trondheim, Norway (email) Abstract: We study stability of solutions of the Cauchy problem on the line for the CamassaHolm equation $u_tu_{xxt}+3uu_x2u_xu_{xx}uu_{xxx}=0$ 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 hyperelasticrod equation $u_tu_{xxt}+f(u)_xf(u)_{xxx}+(g(u)+\frac12 f''(u)(u_x)^2)_x=0$ (for $f$ without inflection points).
Keywords: CamassaHolm equation, Lipschitz
metric, conservative solutions.
Received: March 2012; Revised: May 2012; Available Online: January 2013. 
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