In this paper we prove well-posedness for a measure-valued continuity equation with solution-dependent velocity and flux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of [EHM15a] to settings where the dynamics are driven by interactions. In a forward-Euler-like approach, we construct a time-discretized version of the original problem and employ the results of [EHM15a] as a building block within each subinterval. A limit solution is obtained as the mesh size of the time discretization goes to zero. Moreover, the limit is independent of the specific way of partitioning the time interval [0,T].
This paper is partially based on results presented in [Eve15, Chapter 5], while a number of issues that were still open there, are now resolved.
Keywords: Measure-valued equations, nonlinearities, time discretization, flux boundary condition, mild solutions, particle systems