# American Institute of Mathematical Sciences

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Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations
July  2013, 33(7): 2809-2827. doi: 10.3934/dcds.2013.33.2809

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

 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).
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
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