L¹ stability estimates for n x n conservation laws

Let u_t + f(u)_x = 0 be a strictly hyperbolic n x n system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional Φ = Φ(u, v), equivalent to the L¹ distance, which is "almost decreasing" i.e., Φ(u(t), v(t)) - Φ(u(s), v(s)) ≦ O (ε) · (t - s) for all t > s ≧ 0, for every pair of ε-approximate solutions u, v with small total variation, generated by a wave front tracking algorithm. The small parameter ε 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 L¹ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by a n x n system of conservation laws.