A constructive approach to Euler hydrodynamics for attractive processes. Application to k-step exclusion

We derive by a constructive method the hydrodynamic behavior of attractive processes with irreducible jumps and product invariant measures. Our approach relies on (i) explicit construction of Riemann solutions without assuming convexity, which may lead to contact discontinuities and (ii) a general result which proves that the hydrodynamic limit for Riemann initial profiles implies the same for general initial profiles. The k-step exclusion process provides a simple example. We also give a law of large numbers for the tagged particle in a nearest neighbor asymmetric k-step exclusion process.