Hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries

We consider attractive particle systems in $\Z^d$ with product invariant measures. We prove that when particles are restricted to a subset of $\Z^d$, with birth and death dynamics at the boundaries, the hydrodynamic limit is given by the unique entropy solution of a conservation law, with boundary conditions in the sense of Bardos, Leroux and Nédélec. For the hydrostatic limit between parallel hyperplanes, we prove a multidimensional version of the phase diagram conjectured in \cite{ps}, and show that it is robust with respect to perturbations of the boundaries.

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