We prove a strong law of large numbers for the rescaled asymmetric simple exclusion process. By a coupling procedure we show that the density profile is a weak solution to a first order quasilinear partial differential equation. Moreover, the monotonicity of the process allows us to show that it is the unique solution satisfying the entropy condition. The local equilibrium is then an easy consequence.
"Hydrodynamical Limit for the Asymmetric Simple Exclusion Process." Ann. Probab. 15 (2) 546 - 560, April, 1987. https://doi.org/10.1214/aop/1176992158