Asymmetric attractive zero-range processes with particle destruction at the origin

We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^beta$, where $beta in mathbb{R}$ and $Ninmathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $beta$. Namely, if $beta geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $beta<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.