Liouville type theorem and kinetic formulation for 2 × 2 systems of conservation laws

We study L infinity entropy solutions to 2 & times;2 systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown to characterize all solutions with finite entropy production. Next, we prove a Liouville-type theorem for genuinely nonlinear systems, which is the main result of the paper. This implies in particular that for every finite entropy solution, every point (t,x) is an element of R+ x R\J is of vanishing mean oscillation, where J subset of R+ x R is a set of Hausdorff dimension at most 1.