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

In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space H. Our solutions are conservative, in the sense that the total energy ∫ (u + ux) 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 dtJ(u(t), v(t)) ≤ κ · 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. 1 Introduction The Camassa-Holm equation has the form ut − utxx + 3uux = 2uxuxx + uuxxx . Equivalently, it can be written as a scalar conservation law with an additional integro-differential term: ut + (u /2)x + Px = 0 , (1.1) where P is defined as a convolution: P . = 1 2 e−|x| ∗ ( u + ux 2 ) . (1.2) For the physical motivations of this equation we refer to [CH], [CM1], [CM2], [J]. Earlier results on the existence and uniqueness of solutions can be found in [XZ1], [XZ2]. One can regard (1.1) as an evolution equation on a space of absolutely continuous functions with derivatives ux ∈ L. In the smooth case, differentiating (1.1) w.r.t. x one obtains uxt + uuxx + u 2 x − ( u + ux 2 ) + P = 0 . (1.3)