The effect of regularisations in front propagation phenomena

This project is mainly concerned with existence, stability, and numerical simulation of solutions of certain model equations of applied mathematics. Travelling waves or propagating fronts, which are special solutions that maintain their profile and move with constant speed, will be the focus of particular attention because they are essential in the explanation of some general phenomena. For example, in continuous media (e.g. gases) disturbances might cause an instantaneous change on the macroscopic quantities (e.g. pressure, density, temperature) that propagates through the medium. An example in every day life are the sonic booms produced by aircrafts. These changes are reflected mathematically in the form of discontinuous solutions or shocks. In this project a class of models for which travelling wave solutions that represent the continuous (internal) structure of shocks is considered, namely models that give a description at a microscopic (e.g. molecular) level (kinetic description). Another model of interest describes infiltration processes, such as rain water entering the ground before forming aquifers, a model that has been developed by engineers. Validation of this empirical model also requires a mathematical understanding, moreover, this so-called pseudo-parabolic class of models exhibits some intriguing (and so far largely unexplored) mathematical properties. A first step in the mathematical analysis is the study of special solutions, such as equilibrium solutions or solutions that, intuitively, describe the physical system at rest. However, in this case experiments show that, before equilibrium is reached, the water does not advance in a homogeneous way, but forms finger-like structures. In this context travelling wave solutions represent the (primarily) gravity driven water front displacing air/gas in the porous ground. In both problems stability of the propagating fronts, by means of finding solution components growing strongly under the influence of perturbations, will therefore represent an important aspect of the analysis. Theoretical investigations will be complemented with numerical simulations. The project will also deal with related model pseudo-parabolic equations that appear in different contexts, e.g. modelling of oil-recovery by water in oil-reservoirs and models describing aggregation of populations. Other mathematical features and special solutions, such as pattern formation and periodic steady solutions in the later example, will be subject of study.The research will be mainly carried out at the University of Nottingham within the department of Applied Mathematics. Other mathematicians working in related field would be involved, for example, rigorous mathematical analysts and mathematicians involved in modelling of phenomena such as the described above. The period of the fellowship is of three years, where the main investigator will visit other institutions and receive other scientists. Communication of mathematical results will take place firstly at talks given at other universities and at conferences, and the results will be published in relevant scientific journals.