Hydrodynamical Equation for Attractive Particle Systems on Zᵈ

We prove conservation of local equilibrium, away from the shock, for some attractive asymmetric particle systems on Zᵈ. The method applies to a class of particle processes which includes zero-range and simple exclusion processes. The main point in the proof is to exploit attractiveness. The hydrodynamic equation obtained is a first-order nonlinear partial differential equation which presents shock waves.