Hydrodynamic limit for asymmetric simple exclusion with accelerated boundaries

We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional finite lattice {1, 2, …, N}. The particles can be created/annihilated at the boundaries with given rates. These rates are $L^\infty$ functions of time and are independent of the jump rates in the bulk (cf. Comm. Math. Phys. 310 (2012) 1–24). The boundary dynamics is modified by a factor $N^θ$ with θ > 0. We study the hydrodynamic limit for the particle density profile under the hyperbolic space-time scale. The macroscopic equation is given by (inviscid) Burgers equation with boundary conditions which are characterized by the boundary entropy (C. R. Acad. Sci. Paris 322 (1996) 729–734). A grading scheme is developed to control the formulation of boundary layers on the microscopic level.