L-1 stability estimates for n x n conservation laws

Let u(t) + f (u)(x) = 0 be a strictly hyperbolic n x n system of conservati on laws, each characteristic field being linearly degenerate or genuinely : nonlinear. In this paper Mie explicitly define a functional Phi = Phi (u, u psilon,), 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 paramet er 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 approx imations converge to a unique limit solution, depending Lipschitz continuou sly 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.