  In this paper we consider a class of continuity equations that are
conditioned to stay in general space-time domains, which is formulated as a
continuum limit of interacting particle systems. Firstly, we study the
well-posedness of the solutions and provide examples illustrating that the
stability of solutions is strongly related to the decay of initial data at
infinity. In the second part, we consider the vanishing viscosity approximation
of the system, given with the co-normal boundary data. If the domain is
spatially convex, the limit coincides with the solution of our original system,
giving another interpretation to the equation.
