On Continuity Equations in Space-time Domains

Abstract: 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.