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 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.

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In higher plants, cell-to-cell polar auxin transport (PAT) of the phytohormone auxin, indole-3-acetic acid (IAA), generates maxima and minima that direct growth and development. Although IAA is present in all plant phyla, PAT has only been detected in land plants, the earliest being the Bryophytes. Charophyta, a group of freshwater green algae, are among the first multicellular algae with a land plant-like phenotype and are ancestors to land plants. IAA has been detected in members of Charophyta, but its developmental role and the occurrence of PAT are unknown. We show that naphthylphthalamic acid (NPA)-sensitive PAT occurs in internodal cells of Chara corallina. The relatively high velocity (at least 4–5 cm/h) of auxin transport through the giant (3–5 cm) Chara cells does not occur by simple diffusion and is not sensitive to a specific cytoplasmic streaming inhibitor. The results demonstrate that PAT evolved early in multicellular plant life. The giant Chara cells provide a unique new model system to study PAT, as Chara allows the combining of real-time measurements and mathematical modelling with molecular, developmental, cellular, and electrophysiological studies.

In this paper we prove well-posedness for a measure-valued continuity equation with solutiondependent velocity and flux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of [EHM15a] 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 [EHM15a] 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 [Eve15, Chapter 5], while a number of issues that were still open there, are now resolved.

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