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Abstract: We study stability of solutions of the Cauchy problem on the line for theCamassa-Holm equation $u t-u {xxt}+3uu x-2u xu {xx}-uu {xxx}=0$ with initialdata $u 0$. In particular, we derive a new Lipschitz metric $d \D$ with theproperty that for two solutions $u$ and $v$ of the equation we have$d \Dut,vt\le e^{Ct} d \Du 0,v 0$. The relationship between this metricand the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends tothe generalized hyperelastic-rod equation$u t-u {xxt}+fu x-fu {xxx}+gu+\frac12 f-uu x^2 x=0$ for $f$without inflection points.



Autor: Katrin Grunert, Helge Holden, Xavier Raynaud

Fuente: https://arxiv.org/







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