# Lipschitz metric for the Camassa-Holm equation on the line - Mathematics > Analysis of PDEs

Lipschitz metric for the Camassa-Holm equation on the line - Mathematics > Analysis of PDEs - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

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/