This paper is concerned with the initial value problem for a strictly hyperbolic n x n system of conservation laws in one space dimension: (*) ut + [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, ·) = Stū 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, ·) = Stū. 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.
|Original language||English (US)|
|Journal||Memoirs of the American Mathematical Society|
|State||Published - Jul 1 2000|
All Science Journal Classification (ASJC) codes
- Applied Mathematics