Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

PB - Khayyam Publishing UR - http://hdl.handle.net/1963/3256 U1 - 1445 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Oleinik type estimates and uniqueness for n x n conservation laws JF - J. Differential Equations 156 (1999), no. 1, 26--49 Y1 - 1999 A1 - Alberto Bressan A1 - Paola Goatin AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/3375 U1 - 955 U2 - Mathematics U3 - Functional Analysis and Applications ER -