L 1 Stability Estimates for n x n Conservation Laws

Let u t + f(u) x = 0 be a strictly hyperbolic n n system of conservation laws, each characteristic eld being linearly degenerate or genuinely nonlinear. In this paper we explicitly deene a functional = (u; v), equivalent to the L 1 distance, which is \almost decreasing" i.e. ? u(t); v(t) ? ? u(s); v(s) O(") (t ? s) for all t > s 0; for every couple of "-approximate solutions u; v with small total variation, generated by a wave front tracking algorithm. The small parameter " here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in u and in v. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the L 1 norm. This provides a new proof of the existence of the Standard Riemann Semigroup generated by a n n system of conservation laws.