Hydrodynamics for one-dimensional ASEP in contact with a class of reservoirs

We study the hydrodynamic behaviour of the asymmetric simple exclusion process on the lattice of size $n$. In the bulk, the exclusion dynamics performs rightward flux. At the boundaries, the dynamics is attached to reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are weakened by $n^\theta$ for some $\theta<0$ and (2) the reservoirs that create particles only at the right boundary and annihilate particles only at the left boundary. We prove that the spatial density of particles, under the hyperbolic time scale, evolves with the entropy solution to a scalar conservation law on $[0,1]$ with boundary conditions. The boundary conditions are characterised by the boundary traces at $x=0$ and $x=1$ which take values from $\{0,1\}$.