The talk will be based on a joint work with L. Ambrosio, G. Crippa and A. Figalli. First, some new well-posedness results for continuity and transport equations with weakly differentiable velocity fields will be discussed. These results can be applied to the analysis of a 2 x 2 system of conservation laws in one space dimension known as the chromatography system, leading to global existence and uniqueness results for suitable classes of entropy admissible solutions.