Mild solutions to a measure-valued mass evolution problem with flux boundary conditions

TL;DR

This paper studies the existence, uniqueness, and approximation of measure-valued solutions to a linear transport equation with flux boundary conditions, using boundary layer analysis and semigroup methods.

Contribution

It introduces a new framework for mild solutions in measure spaces and provides convergence rates for approximation methods with boundary layer insights.

Findings

Established well-posedness of measure-valued solutions.

Derived convergence rates for approximation procedures.

Analyzed flux boundary conditions in the singular limit.

Abstract

We investigate the well-posedness and approximation of mild solutions to a class of linear transport equations on the unit interval $[0,1]$ endowed with a linear discontinuous production term, formulated in the space $\mathcal{M}([0,1])$ of finite Borel measures. Our working technique includes a detailed boundary layer analysis in terms of a semigroup representation of solutions in spaces of measures able to cope with the passage to the singular limit where thickness of the layer vanishes. We obtain not only a suitable concept of solutions to the chosen measure-valued evolution problem, but also derive convergence rates for the approximation procedure and get insight in the structure of flux boundary conditions for the limit problem.