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

Alberto Bressan·Massimo Fonte
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Abstract

In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space $H^1$. Our solutions are conservative, in the sense that the total energy $\int (u^2+u_x^2) dx$ 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 ${d\over dt} J(u(t), v(t))\leq κ\cdot J(u(t),v(t))$ 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. 2005-05-04. An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation. https://arxiv.org/abs/math/0504450

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