On some partial differential equation models in socio-economic contexts - analysis and numerical simulations

Abstract

This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry and Lions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented. The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension, the dynamic behaviour of solutions of the original model is explored for some special cases. These results are compared to numerical simulations. Moreover, we consider a discrete cellular automaton model introduced by A. Kirchner and A. Schadschneider in 2002. By (formally) passing to the continuum limit, we obtain a system of partial differential equations. Some analytical properties, such as linear stability of stationary states, are examined and extensive numerical simulations show capabilities and limitations of the model in both the discrete and continuous setting.