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

TL;DR

This paper establishes well-posedness for a measure-valued continuity equation with solution-dependent velocities and flux boundary conditions, extending previous work to include interaction-driven dynamics and proving independence of discretization methods.

Contribution

It generalizes prior results to interaction-driven dynamics and demonstrates the existence and uniqueness of solutions via a time-discretization approach.

Findings

Well-posedness of the measure-valued problem is proven.

Limit solutions are independent of time partitioning.

The approach extends previous models to include interactions.

Abstract

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 [Evers, Hille and Muntean. Journal of Differential Equations, 259:1068-1097, 2015] 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 [Evers, Hille and Muntean. Journal of Differential Equations, 259:1068-1097, 2015] 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 [Evers. PhD thesis, Eindhoven University of Technology, 2015], while a number of issues that were still open there, are now resolved.