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In this paper we prove well-posedness for a measure-valued continuityequation with solution-dependent velocity and flux boundary conditions, posedon a bounded one-dimensional domain. We generalize the results of [Evers, Hilleand Muntean. Journal of Differential Equations, 259:1068-1097, 2015] tosettings where the dynamics are driven by interactions. In a forward-Euler-likeapproach, we construct a time-discretized version of the original problem andemploy the results of [Evers, Hille and Muntean. Journal of DifferentialEquations, 259:1068-1097, 2015] as a building block within each subinterval. Alimit solution is obtained as the mesh size of the time discretization goes tozero. Moreover, the limit is independent of the specific way of partitioningthe time interval $[0,T]$. This paper is partially based on results presentedin Chapter 5 of [Evers. PhD thesis, Eindhoven University of Technology, 2015],while a number of issues that were still open there, are now resolved.

Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities