%0 Report %D 2005 %T An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation %A Alberto Bressan %A Massimo Fonte %X 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 + u2x) 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. %B Methods Appl. Anal. 12 (2005) 191-219 %G en_US %U http://hdl.handle.net/1963/1719 %1 2432 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-01-24T09:03:54Z\\nNo. of bitstreams: 1\\nmath.AP0504450.pdf: 261370 bytes, checksum: 75945d031343a82836b46ab9705ed6de (MD5) %0 Journal Article %J J. Differential Equations 156 (1999), no. 1, 26--49 %D 1999 %T Oleinik type estimates and uniqueness for n x n conservation laws %A Alberto Bressan %A Paola Goatin %X Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleĭnik in the scalar case. %B J. Differential Equations 156 (1999), no. 1, 26--49 %I Elsevier %G en_US %U http://hdl.handle.net/1963/3375 %1 955 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-01T09:26:02Z\\nNo. of bitstreams: 1\\nOleinik_type.pdf: 1567166 bytes, checksum: ba588e7f2b587d26f5b613a53557bb2f (MD5) %R 10.1006/jdeq.1998.3606