Hydrodynamics of a driven lattice gas with open boundaries: the   asymmetric simple exclusion
  We consider the asymmetric simple exclusion process in $d\ge 3$ with open boundaries. The particle reservoirs of constant densities are modeled by birth and death processes at the boundary. We prove that, if the initial density and the densities of the boundary reservoirs differ for order of $\epsilon$ from 1/2, the density empirical field, rescaled as $\epsilon^{-1}$, converges to the solution of the initial-boundary value problem for the viscous Burgers equation in a finite domain with given density on the boundary. 
O. Benois, R. Esposito, R. Marra and M. Mourragui