  Well-posedness for the two dimensional Euler system with given initial
vorticity is known since the works of Judovi\v{c}. In this paper we show
existence of solutions in the case where we allowed the fluid to enter in and
exit from the boundaries and from some points of the fluid domain. In
particular we derive the equations of the model as the limit when we replace
the points by some small holes. To do that we extend the DiPerna-Lions theory
with non-tangent velocity field on the boundary to the case of time-dependent
domain, we extend the existence result for the two dimensional Euler system
with in-out flow to time-dependent domain and finally we derive the system that
models a fluid which is allowed to enter in and exit from the boundary and some
points. The solutions are characterized by the presence of source, sink and
vortex points.
