TY  - GEN
ID  - heidok35450
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/35450/
A1  - Düll, Christian Alexander
AV  - public
CY  - Heidelberg
N2  - This thesis deals with Radon measures and how they can be used to extend nonlinear structured population models from the classical Euclidean state spaces to abstract Polish metric spaces. To this end, we first investigate the functional analytic properties of the space of Radon measures under the flat norm and show that in some sense the latter generalises the well-known Wasserstein distance W_1 from the conservative to the unbalanced case. We then apply variational inequality theory to derive an explicit formula for the flat distance between a linear combination of Dirac deltas and a fixed Dirac measure.  

The second part of the thesis deals with structured population models in measures. As a start, we consider the Euclidean case and prove well-posedness of the linear model, first without and then with state-independent influx, using Duhamel's principle. Subsequently, the nonlinear model is solved via a reduction to a linear model with frozen measure arguments. 
The gathered insights in the R^d case enable us to abstract the necessary concepts so that we can transfer the model formulation to general Polish metric spaces. However, in the absence of a vector space structure, we cannot rely on a governing differential equation, and  introduce a suitable implicit integral representation instead which serves both as model and notion of solution. A major step towards well-posedness of the involved Bochner integrals is given by the structure of the space of measures and its favorable properties under the flat norm.

We conclude with various outlooks  on the applicability of our theory. At first, several ideas for promising models in measures  that can fully exploit the abstract  Polish state spaces are sketched. Afterwards, we show how to incorporate measure differential equations and - with a minimal adaptation- also the class of  coagulation-fragmentation models into our framework. Finally, it is  proven that our flat norm is closely related to a transport type distance developed by Fournier and Perthame to study the asymptotics of nonexpanding transport processes.
Y1  - 2024///
TI  - Generalising nonlinear population models-
Radon measures, Polish spaces and the flat norm
ER  -