Lipschitz metric for the Camassa-Holm equation on the line

Abstract

We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ 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 the usual norms in $H^1$ and $L^{\infty }$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u{)}_x-f(u{)}_{xxx}+(g(u)+\frac{1}{2}f"(u)(u_x{)}^2{)}_x=0$ (for $f$ without inflection points).

References in corpus (2)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Lipschitz metric for the periodic Camassa-Holm equation

Cited by in corpus (14)

• Global solutions for the two-component Camassa-Holm system
• Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics
• Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics
• The general peakon-antipeakon solution for the Camassa-Holm equation
• Existence and Lipschitz stability for $\alpha$-dissipative solutions of the two-component Hunter-Saxton system
• Lipschitz Stability for the Hunter-Saxton Equation
• On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system
• Lipschitz metric for the two-component Camassa--Holm system
• Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation
• A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system
• Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics
• Unique Continuation Properties for solutions to the Camassa-Holm equation and other non-local equations
• Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $\lambda$-family equations
• A Lipschitz metric for the Camassa--Holm equation