Preprint 2005-018

An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation

Alberto Bressan and Massimo Fonte

Abstract: In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space $H1$. Our solutions are conservative, in the sense that the total energy $\int (u2+u_x2)\, dx$ remains a.e.~constant in time. Our new approach is based on a distance functional $J(u,v)$, defined in terms of an optimal transportation problem, which satisfies ${d\over dt} J(u(t), v(t))\leq \kappa\cdot J(u(t),v(t))$ for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result.

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Alberto Bressan, <>
Massimo Fonte
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<> April 20 2005.

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Last modified: Thu Apr 21 09:45:08 MEST 2005