Solutions to a measured-valued mass evolution problem with flux boundary conditions inspired by crowd dynamics

Motivated by our research on pedestrian flows, we study a non-conservative measure-valued evolution problem posed in a finite interval and explore the possibility of imposing a flux boundary condition. The main steps of our work include the analysis of a suitably scaled regularized problem possessing a boundary layer that accumulates mass and detailed investigations of the boundary layer by means of semigroup techniques in spaces of measures. We consider passage to the singular limit where thickness of the layer vanishes (resembling the fast reaction asymptotics typical for systems with slow transport and rapid reactions). We obtain not only suitable solutions to the measured-value evolution problem, but also derive a convergence rate for the approximation procedure as well as the structure of (flux) boundary conditions for the limit problem.