  In this paper, a diffusion-aggregation equation with delta potential is
introduced. Based on the global existence and uniform estimates of solutions to
the diffusion-aggregation equation, we also provide the rigorous derivation
from a stochastic particle system while introducing an intermediate particle
system with smooth interaction potential. The theoretical results are compared
to numerical simulations relying on suitable discretization schemes for the
microscopic and macroscopic level. In particular, the regime switch where the
analytic theory fails is numerically analyzed very carefully and allows for a
better understanding of the equation.
