Beschreibung
In this thesis a framework for modelling two coupled hypersurfaces evolving in a three-dimensional domain that is filled with a fluid is developed in terms of partial differential equations that can be classified as free boundary problems. In particular, the coupling of both hypersurfaces depends on particles of a species that evolve on one of the surfaces. Since they influence the surface tension, they can be understood as surfactants. We use two common approaches for modelling free boundary problems, namely sharp and diffuse interfaces by using similar physical arguments for their derivation as well as a formal asymptotic analysis. From the modelling framework, we specify particular models for the evolution of biomembranes, and, even more specific, for studying the biological phenomenon of cell blebbing. Properties of these models are investigated by means of rigorous mathematical analysis showing existence of solutions, existence and stability of stationary solutions where possible, as well a singular limits. We also present numerical algorithms for simulative studies of cell blebbing models and report and interpret results obtained from their computations. Thus we can validate that our models qualitatively reproduce phenomena that have been reported in the biological literature.
