Hyperbolic hydrodynamic limits for interacting particle systems with several conservation laws - survey and outlook

The typical and much studied model of interacting particles with one single conservation law is the asymmetric simple exclusion which leads, under Eulerian scaling, to the Burgers equation as hydrodynamic limit. This one, and similar settings have been fully understood since the 1990-s. When there are more than one conserved observables (like, e.g., several types of interacting particles, or particle number and a kind of momentum) the limit and the limiting procedure are much more complex. Hyperbolic systems of conservation laws arise and the status of the limiting procedure is less understood.

I will survey old work in this context (joint with Benedek Valkó and József Fritz) and flash up some recent projects in a very early stage (joint with Balázs Maga).