Maximal dissipation in hyperbolic equations - how to select physical solutions?

We first discuss a natural criterion for selection of unique solutions of hyperbolic PDEs - the so-called maximal dissipation criterion. Then we show how to apply it to the so-called Hunter-Saxton equation, known from the theory of liquid crystals. Our result (joint work with T. Cieslak), based on considerable extension of methods introduced by Dafermos, seems to be a first result, where the criterion has been effectively applied within general class of weak solutions.