Hyperbolic pdes ( systems of conservation law) is a dead subject. I think Bressan and his group has solved most big problems in the field. From what people told me, Bressan has stopped doing hyperbolic pdes and is doing pdes in math bio now. Regarding PDEs, any top 10 schools except harvard has a strong group in pdes. However, i don't know if anyone is still doing hyperbolic pdes.