  Recently, Grabowska and Kaplan suggested a non-perturbative formulation of a
chiral gauge theory, which consists of the conventional domain-wall fermion and
a gauge field that evolves by the gradient flow from one domain wall to the
other. In this paper, we discuss the U(1) axial-vector current in 4 dimensions
using this formulation. We introduce two sets of domain-wall fermions belonging
to complex conjugate representations so that the effective theory is a
4-dimensional vector-like gauge theory. Then, as a natural definition of the
axial-vector current, we consider a current that generates the simultaneous
phase transformations for the massless modes in 4 dimensions. However, this
current is exactly conserved and does not reproduce the correct anomaly. In
order to investigate this point precisely, we consider the mechanism of the
conservation. We find that this current includes not only the axial current on
the domain wall but also a contribution from the bulk, which is non-local in
the sense of 4-dimensional fields. Therefore, the local current is obtained by
subtracting the bulk contribution from it.
