Lyapunov Functional for Solutions of Systems of Conservation Laws Containing a Strong Rarefaction

We study the Cauchy problem for a strictly hyperbolic n×n system of conservation laws in one space dimension: ut + f(u)x = 0, u(0, x) = ū(x). The initial data ū is a small BV perturbation of a single rarefaction wave with an arbitrary strength. All characteristic fields are assumed to be genuinely nonlinear or linearly degenerate in the vicinity of the reference rarefaction curve. We prove that a suitable BV stability condition yields uniform bounds on the total variation of perturbation, thus implying the existence of a global admissible solution. On the other hand, a stronger L1 stability condition guarantees the existence of the Lipschitz continuous flow of solutions. Our proof relies on the construction of a Lyapunov functional which is almost decreasing in time and which is equivalent to the L1 distance between the two solutions.