Lipschitz metric for the Camassa-Holm equation on the line

By Katrin Grunert, Helge Holden, Xavier Raynaud. 2010

47 citations recorded by the source corpus.

We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ 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 the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f"(u)(u_x)^2)_x=0$ (for $f$ without inflection points).

Subfields: Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4c8ce931ac6e1a1b675c34f72c622cbe729bac98

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In Mathematical Physics

In Modeling and Simulation

In Statistical and Nonlinear Physics

Citations in the corpus

Generic regularity of conservative solutions to a nonlinear wave equation (2015, methodology): “A detailed analysis of relabeling symmetries, in connection with the Camassa-Hom equation, can be found in [13].”
Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics (2014, background): “The papers [4, 8, 9] have introduced new distances d♦(·, ·) on H1 which render the flow map uniformly Lipschitz continuous on bounded subsets of H1.”