Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities (Q2811886)

The author summarize the contents of this paper in the abstract of the paper as follows: 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 an earlier paper [the authors, J. Differ. Equations 259, No. 3, 1068--1097 (2015; Zbl 1315.35057)] 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 those results 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 Chapter 5 of [the first author, Evolution equations for systems governed by social interactions. Eindhoven: Eindhoven University of Technology (PhD Thesis) (2015)], while a number of issues that were still open there are now resolved.