Lipschitz metric for the periodic Camassa-Holm equation

Abstract

We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa u_x+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ 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 $H_{\mathrm{per}}^1$ and $L_{\mathrm{per}}^{\infty }$ is clarified.

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• Global Lagrangian solutions of the Camassa-Holm equation
• Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics
• Lipschitz Metrics for a Class of Nonlinear Wave Equations
• Lipschitz metric for the Novikov equation
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• Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $\lambda$-family equations