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 x n system of conservation laws in one space dimension: (*) u_t + [F(u)]_x = 0, u(0, x) = ū(x). Each characteristic field is assumed to be either linearly degenerate or genuinely nonlinear. We prove that there exist a domain D ⊂ L¹, containing all functions with sufficiently small total variation, and a uniformly Lipschitz continuous semigroup S: D x [0, ∞[→ D with the following properties. Every trajectory t → u(t, ·) = S_tū 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 ∈ [0, T], then it coincides with the semigroup trajectory, i.e. u(t, ·) = S_tū. For a given domain D, the semigroup 5 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 x n systems of conservation laws, with small initial data.