Uniqueness of Weak Solutions to Systems of Conservation Laws

. Consider a strictly hyperbolic n \Theta n system of conservation laws in one space dimension: u t + F (u) x = 0: () Relying on the existence of the Standard Riemann Semigroup generated by (), we establish the uniqueness of entropy-admissible weak solutions to the Cauchy problem, under a mild assumption on the variation of u along space-like segments. 1 - Introduction. We are concerned with the uniqueness of solutions to the Cauchy problem for an hyperbolic n \Theta n system of conservation laws in one space dimension: u t + F (u) x = 0; (1:1) u(0; x) = ¯ u(x): (1:2) Let\Omega ` IR n be an open set containing the origin, and let F :\Omega 7! IR n be a smooth map. Assume that the system (1.1) is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear [12, 18]. For every initial data ¯ u 2 L 1 with small total variation, a well known theorem of Glimm [10] provides the global existence of weak solutions. In subsequent papers..