Lipschitz metric for the periodic Camassa-Holm equation

Details Lipschitz metric for the periodic Camassa-Holm equation Lipschitz metric for the periodic Camassa-Holm equation

2010-05-19

arxiv.org/abs/1005.3440v1

Mathematics

Analysis of PDEs

Scientific paper

We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ 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 usual norms in $H^1_{\rm per}$ and $L^\infty_{\rm per}$ is clarified.

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