  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.
