Nachwuchsgruppe / Junior Research Group
Partielle Differentialgleichungen und Anwendungen (01.10.2012-30.09.2014)
The research group investigates linear and nonlinear partial differential equations with surface and interface conditions from a mathematical point of view. Our focus lies on parabolic equations of heat equation, porous medium or reaction-diffusion type.
In many real-world situations one considers reaction-diffusion processes in a thin three-dimensional layer. As the thickness of the surface tends to zero in a so-called singular limit one has to deal with different scalings. According to these one can derive limit models on a two dimensional surface, which should still feature the main properties of the thin layer model which are of interest. To make such a dimension reduction rigorous one formulates the orginal model and the limit model within the same mathematical framework and performs the singular limit appropriately within this framework. Here the difficulties arise especially from nonlinear diffusion and from the coupling of reactions for different species. Another reason to study singular limits in the context of surface processes are fast reaction limits, i.e., when reactions in the bulk and on the surface live on different time scales, and further almost vanishing diffusion coefficients.
As a next step one analyses the limit models with nonlinear surface and/or interface conditions. Here our main interests are existence and uniqueness of solutions, their regularity and their qualitative behavior, with focus on the interplay of bulk, surface and interface processes. In particular we investigate global-in-time existence of solutions, control theory in terms of observability and the spectral theory of interface problems. If possible we allow for realistic non-smooth scenarios, i.e., coefficients with jumps and bulks/surfaces with corners.
(letzte Änderung: 23.02.2021, 08:27 Uhr)