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 Mie explicitly define a functional Phi = Phi (u, upsilon,), equivalent to the L-1 distance, which is "almost decreasing" i.e., Phi(u (t), upsilon (t)) - Phi(u (s), upsilon (s)) less than or equal to O (epsilon) . (t - s) for all t > s greater than or equal to 0, for every pair of epsilon-approximate solutions u, upsilon with small total variation, generated by a wave front tracking algorithm. The small parameter epsilon here controls he errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all nonphysical waves in u and in upsilon. 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: x n system of conservation laws.