Well-posedness of the Cauchy Problem for N X N Systems of Conservation Laws

This paper is concerned with the initial value problem for a strictly hyperbolic n n system of conservation laws in one space dimension: u t + F(u) x = 0; u(0; x) = u(x): () Each characteristic eld is assumed to be either linearly degenerate or genuinely nonlinear. We prove that there exists a domain D L 1 , containing all functions with suuciently small total variation, and a uniformly Lipschitz continuous semigroup S : D 0; 1 7 ! D with the following properties. Every trajectory t 7 ! u(t;) = S t u of the semigroup is a weak, entropy-admissible solution of (). Viceversa, if a piecewise Lipschitz, entropic solution u = u(t; x) of () exists for t 2 0; T], then it coincides with the semigroup trajectory, i.e. u(t;) = S t u. For a given domain D, the semigroup S with the above properties is unique. These results yield the uniqueness, continuous dependence and global stability of weak, entropy-admissible solutions of the Cauchy problem (), for general n n systems of conservation laws, with small initial data.