An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation
Alberto Bressan, Massimo Fonte
TL;DR
The paper addresses constructing a global, continuous flow of conservative solutions to the Camassa-Holm equation on the entire space $H^1$, with the total energy $\int (u^2+u_x^2) dx$ remaining almost everywhere constant in time. It introduces a novel distance functional $J(u,v)$ defined via an optimal transportation problem, for which $\frac{d}{dt}\,J(u(t),v(t)) \le \kappa \cdot J(u(t),v(t))$ along solution pairs. Using this metric framework, arbitrary solutions can be obtained as the uniform limit of multi-peakon solutions and a general uniqueness result follows. The approach provides a global, continuous flow on $H^1$ with a rigorous metric structure that facilitates analysis of Camassa-Holm dynamics and their approximation.
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 ${d\over dt} J(u(t), v(t))\leq \kappa\cdot J(u(t),v(t))$ 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.