  We studied the continuity equation in presence of a local potential, and a
non-local potential arising from electron-electron interaction in both
commutative and non-commutative phase-space. Furthermore, we examined the
influence of the phase-space non-commutativity on both the locality and the
non-locality, where the definition of current density in commutative
phase-space cannot satisfy the condition of current conservation, but with the
steady state, in order to solve this problem, we give a new definition of the
current density including the contribution due to the non-local potential. We
showed that the calculated current based on the new definition of current
density maintains the current. As well for the case when the non-commutativity
in phase-space considered, we found that the conservation of the current
density completely violated; and the non-commutativity is not suitable for
describing the current density in presence of non-local and local potentials.
Nevertheless, under some conditions, we modified the current density to solve
this problem. Subsequently, as an application we studied the Frahn-Lemmer
non-local potential, taking into account that the employed methods concerning
the phase-space non-commutativity are both of Bopp-shift linear transformation
through the Heisenberg-like commutation relations, and the Moyal-Weyl product.
