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

We study L∞ entropy solutions to 2×2 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)∈R+×R∖J is of vanishing mean oscillation, where J⊂R+×R is a set of Hausdorff dimension at most 1.